Comparative genomics reveals birth and death of fragile regions in mammalian evolution
 Max A Alekseyev^{1}Email author and
 Pavel A Pevzner^{2}Email author
DOI: 10.1186/gb20101111r117
© Alekseyev et al.; licensee BioMed Central Ltd. 2010
Received: 15 July 2010
Accepted: 30 November 2010
Published: 30 November 2010
Abstract
Background
An important question in genome evolution is whether there exist fragile regions (rearrangement hotspots) where chromosomal rearrangements are happening over and over again. Although nearly all recent studies supported the existence of fragile regions in mammalian genomes, the most comprehensive phylogenomic study of mammals raised some doubts about their existence.
Results
Here we demonstrate that fragile regions are subject to a birth and death process, implying that fragility has a limited evolutionary lifespan.
Conclusions
This finding implies that fragile regions migrate to different locations in different mammals, explaining why there exist only a few chromosomal breakpoints shared between different lineages. The birth and death of fragile regions as a phenomenon reinforces the hypothesis that rearrangements are promoted by matching segmental duplications and suggests putative locations of the currently active fragile regions in the human genome.
Background
In 1970 Susumu Ohno [1] came up with the Random Breakage Model (RBM) of chromosome evolution, implying that there are no rearrangement hotspots in mammalian genomes. In 1984 Nadeau and Taylor [2] laid the statistical foundations of RBM and demonstrated that it was consistent with the human and mouse chromosomal architectures. In the next two decades, numerous studies with progressively increasing resolution made RBM the de facto theory of chromosome evolution.
RBM was refuted by Pevzner and Tesler [3] who suggested the Fragile Breakage Model (FBM) postulating that mammalian genomes are mosaics of fragile and solid regions. In contrast to RBM, FBM postulates that rearrangements are mainly happening in fragile regions forming only a small portion of the mammalian genomes. While the rebuttal of RBM caused a controversy [4–6], Peng et al. [7] and Alekseyev and Pevzner [8] revealed some flaws in the arguments against FBM. Furthermore, the rebuttal of RBM was followed by many studies supporting FBM [9–31].
Comparative analysis of the human chromosomes reveals many short adjacent regions corresponding to parts of several mouse chromosomes [32]. While such a surprising arrangement of synteny blocks points to potential rearrangement hotspots, it remains unclear whether these regions reflect genome rearrangements or duplications/assembly errors/alignment artifacts. Early studies of genomic architectures were unable to distinguish short synteny blocks from artifacts and thus were limited to constructing large synteny blocks. Ma et al. [33] addressed the challenge of constructing highresolution synteny blocks via the analysis of multiple genomes. Remarkably, their analysis suggests that there is limited breakpoint reuse, an argument against FBM, that led to a split among researchers studying chromosome evolution and raised a challenge of reconciling these contradictory results. Ma et al. [33] wrote: 'a careful analysis [of the RBM vs FBM controversy] is beyond the scope of this study' leaving the question of interpreting their findings open. Various models of chromosome evolution imply various statistics and thus can be verified by various tests. For example, RBM implies exponential distribution of the synteny block sizes, consistent with the humanmouse synteny blocks observed in [2]. Pevzner and Tesler [3] introduced the 'pairwise breakpoint reuse' test and demonstrated that while RBM implies low breakpoint reuse, the humanmouse synteny blocks expose rampant breakpoint reuse. Thus RBM is consistent with the 'exponential length distribution' test [2] but inconsistent with the 'pairwise breakpoint reuse' test [34]. Both these tests are applied to pairs of genomes, not taking an advantage of multiple genomes that were recently sequenced. Below we introduce the 'multispecies breakpoint reuse' test and demonstrate that both RBM and FBM do not pass this test. We further propose the Turnover Fragile Breakage Model (TFBM) that extends FBM and complies with the multispecies breakpoint reuse test.
Technically, findings in [33] (limited breakpoint reuse between different lineages) are not in conflict with findings in [3] (rampant breakpoint reuse in chromosome evolution). Indeed, Ma et al. [33] only considered reuse between different branches of the phylogenetic tree (interreuse) and did not analyze reuse within individual branches (intrareuse) of the tree. TFBM reconciles the recent studies supporting FBM with the Ma et al. [33] analysis. We demonstrate that data in [33] reveal rampant but elusive breakpoint reuse that cannot be detected via counting repeated breakages between various pairs of branches of the evolutionary tree. TFBM is an extension of FBM that reconciles seemingly contradictory results in [9–31] and [33] and explains that they do not contradict to each other. TFBM postulates that fragile regions have a limited lifespan and implies that they can migrate between different genomic locations. The intriguing implication of TFBM is that few regions in a genome are fragile at any given time raising a question of finding the currently active fragile regions in the human genome.
While many authors have discussed the causes of fragility, the question what makes certain regions fragile remains open. Previous studies attributed fragile regions to segmental duplications [35–38], high repeat density [39], high recombination rate [40], pairs of tRNA genes [41, 42], inhomogeneity of gene distribution [7], and long regulatory regions [7, 17, 26]. Since we observed the birth and death of fragile regions, we are particularly interested in features that are also subject to birth and death process. Recently, Zhao and Bourque [38] provided a new insight into association of rearrangements with segmental duplications by demonstrating that many rearrangements are flanked by Matching Segmental Duplications (MSDs), that is, a pair of long similar regions located within a pair of breakpoint regions corresponding to a rearrangement event. MSDs arguably represent an ideal match for TFBM among the features that were previously implicated in breakpoint reuses. TFBM is consistent with the hypothesis that MSDs promote fragility since the similarity between MSDs deteriorates with time, implying that MSDs are also subjects to a 'birth and death' process.
Results and Discussion
Rearrangements and breakpoint graphs
Let P and Q be 'black' and 'red' genomes on the same set of synteny blocks X. The breakpoint graph G(P, Q ) is defined on the set of vertices V = {x^{ t } , x^{ h }  x ∈ χ} with black and red edges inherited from genomes P and Q (Figure 1b). The black and red edges form a collection of alternating blackred cycles in G(P, Q ) and play an important role in analyzing rearrangements (see [45] for background information on genome rearrangements). The trivial cycles in G(P, Q), formed by pairs of parallel black and red edges, represent common adjacencies between synteny blocks in genomes P and Q. Vertices of the nontrivial cycles in G(P, Q) represent breakpoints that partition genomes P and Q into (P, Q)synteny blocks (Figure 1c). The 2break distance d(P, Q) between circular genomes P and Q is defined as the minimum number of 2breaks required to transform one genome into the other (Figure 1d). In contrast to the genomic distance [46] (for linear genomes), the 2break distance for circular genomes is easy to compute [47]:
Theorem 1 The 2break distance between circular genomes P and Q is d(P, Q) = b(P, Q)  c(P, Q), where b(P, Q ) and c(P, Q) are respectively the number of (P, Q)synteny blocks and nontrivial blackred cycles in G(P, Q).
Inter and intrabreakpoint reuse
Given simulated data, one can compute br(e) for all branches and br(e_{1}, e_{2} ) for all pairs of branches in the phylogenetic tree. However, for real data, rearrangements along the branches are unknown, calling for alternative ways for estimating the inter and intrareuse.
Cycles in the breakpoint graphs provide yet another way to estimate the inter and intrareuse. For a branch e = (P, Q) of the phylogenetic tree, one can estimate br(e) by comparing the 2break distance d(P, Q ) and the number of breakpoints 2 · b(P, Q) between the genomes P and Q. This results in the lower bound bound(e) = 4 · d(P, Q) 2 · b(P, Q) for BR(e) [34] that also gives a good approximation for br(e ). On the other hand, one can estimate br(e_{1}, e_{2}) as the number bound(e_{1}, e_{2}) of vertices shared between nontrivial cycles in the breakpoint graphs corresponding to the branches e_{1} and e_{2} (similar approach was used in [48] and later explored in [12, 33]). Assuming that the genomes at the internal nodes of the phylogenetic tree can be reliably reconstructed [33, 49–51], one can compute bound(e) and bound(e_{1}, e_{2}) for all (pairs of) branches. Below we show that these bounds accurately approximate the intra and interreuse.
Analyzing breakpoint reuse (simulated genomes)
We start from analyzing simulated data based on FBM with n fragile regions present in k genomes that evolved according to a certain phylogenetic tree (for the varying parameter n ). We represent one of the leaf genomes as the genome with 20 random circular chromosomes and simulate hundred 2breaks on each branch of the tree.
Below we describe analytical approximations for the values in table in Figure 5. Since every 2break uses four out of 2n vertices in the genome graph, a random 2break uses a vertex υ with the probability $\frac{2}{n}$. Thus, a sequence of t random 2breaks does not use a vertex υ with the probability ${(1\frac{2}{n})}^{t}\approx {e}^{\frac{2t}{n}}(for\phantom{\rule{0.5em}{0ex}}t\ll n)$. For branches e_{1} and e_{2} with respectively t_{1} and t_{2} random 2breaks, the probability that a particular vertex is interreused on e_{1} and e_{2} is approximated as $(1{e}^{\frac{2{t}_{1}}{n}})\cdot (1{e}^{\frac{2{t}_{2}}{n}})$. Therefore, the expected number of interreused vertices is approximated as $2n\cdot (1{e}^{\frac{2{t}_{1}}{n}})\cdot (1{e}^{\frac{2{t}_{2}}{n}})$. Below we will compare the observed interreuse with the expected interreuse in FBM to see whether they are similar thus checking whether FBM represents a reasonable null hypothesis. We will use the term scaled interreuse to refer to the observed interreuse divided by the expected interreuse. If FBM is an adequate null hypothesis we expect the scaled interreuse to be close to one.
Similarly, a sequence of t random 2breaks uses a vertex υ exactly once with the probability $t\cdot \frac{2}{n}\cdot {\left(1\frac{2}{n}\right)}^{t1}\approx \frac{2t}{n}{e}^{\frac{2(t1)}{n}}$. Therefore, the probability of a particular vertex being intrareused on a branch with t random 2breaks is approximately $1{e}^{\frac{2t}{n}}\frac{2t}{n}{e}^{\frac{2(t1)}{n}}$, implying that the expected intrareuse is approximately $2n\cdot \left(1{e}^{\frac{2t}{n}}\frac{2t}{n}{e}^{\frac{2(t1)}{n}}\right)$. We will use the term scaled intrareuse to refer to the observed n^{ e } intrareuse divided by the expected intrareuse. Table S1 in Additional file 1 shows the scaled intra and interreuse for 21 pairs of branches (averaged over 100 simulations) and illustrates that they all are close to one.
In the case when the branch lengths vary, we find it convenient to represent data in Table S2 in Additional file 1 in a different way (as a plot) that better illustrates variability in the scaled interuse. We define the distance between branches e_{1} and e_{2} in the phylogenetic tree as the distance between their midpoints, that is, the overall length of the path, starting at e_{1} and ending at e_{2}, minus $\frac{d({e}_{1})+d({e}_{2})}{2}$. For example, $d(M+,H+)=56+170+58+28\frac{56+28}{2}=270$ (see Figure 4). The xaxis in Figure S1 in Additional file 1, 2 represents the distances between pairs of branches (21 pairs total), while yaxis represents the scaled interreuse for pairs of branches at the distance x.
Surprising irregularities in breakpoint reuse in mammalian genomes
The branch lengths shown in Figure 4 actually represent the approximate numbers of rearrangements on the branches of the phylogenetic tree for Mouse, Rat, Dog, macaQue, and Human genomes (represented in the alphabet of 433 'large' synteny blocks exceeding 500, 000 nucleotides in human genome [50]). For the mammalian genomes, M, R, D, Q, and H, we first used MGRA [50] to reconstruct genomes of their common ancestors (denoted MR, MRD, and QH in Figure 4) and further estimated the breakpoint interreuse between pairs of branches of the phylogenetic tree. The resulting table in Figure 7 reveals some striking differences from the simulated data (Figure 6) that follow a peculiar pattern: the larger is the distance between two branches, the smaller is the amount of interreuse between them (in contrast to RBM/FBM where the amount of interreuse does not depend on the distance between branches). The statement above is imprecise since we have not described yet how to compare the amount of interreuse for different branches at various distances. However, we can already illustrate this phenomenon by considering branches of similar length that presumably influence the interreuse in a similar way (see below).
Below we modify FBM to come up with a new model of chromosome evolution, explaining the surprising irregularities in the interreuse across mammalian genomes.
Turnover fragile breakage model: birth and death of fragile regions
We start with a simulation of 100 rearrangements on every branch of the tree in Figure 4. However, instead of assuming that fragile regions are fixed, we assume that after every rearrangement x fragile regions 'die' and x fragile regions are 'born' (keeping a constant number of fragile regions throughout the simulation). We assume that the genome has m potentially 'breakable' sites but only n of them are currently fragile (n ≤ m) (the remaining n  m sites are currently solid). The dying regions are randomly selected from n currently fragile regions, while the newly born regions are randomly selected from m  n solid regions. The simplest TFBM with a fixed rate of the 'birth and death' process is defined by the parameters m, n, and turnover rate x. FBM is a particular case of TFBM corresponding to x = 0 and n <m, while RBM is a particular case of TFBM corresponding to x = 0 and n = m. While this oversimplistic model with a fixed turnover rate may not adequately describe the real rearrangement process, it allows one to analyze the general trends and to compare them to the trends observed in real data. We further remark that the goal of this paper is to develop a test for distinguishing between TFBM and FBM/RBM rather than a test for distinguishing between FBM and RBM. Thus, our simulations do not distinguish between FBM (x = 0 and n <m) and RBM (x = 0 and n = m) since they do not affect m  n inactive breakpoints in FBM. To distinguish FBM from RBM, one has to analyze the long cycles in the breakpoint graph and the distribution of synteny block sizes (see [3, 8]).
Figure 10 shows the scaled interreuse averaged over yellow, green, and red cells that reveals a different behavior between FBM and TFBM. Indeed, while the scaled interreuse is close to 1 for all pairs of branches in the case of FBM, it varies in the case of TFBM. For example, for n = 900, m = 2, 000, and x = 3, the interreuse in yellow cells is approximately 40, in green cells is approximately 45, and in red cells is approximately 56. Table S3 in Additional file 1 presents the differences in the interreuse between red, green, and yellow cells as a function of m and x (for n = 900). In Methods we describe a formula for estimating the breakpoint interreuse in the case of TFBM that accurately approximates the values shown in Figure 10.
Multispecies breakpoint reuse test
that represents the total breakpoint reuse between pairs of rearrangements ρ_{1}, ρ_{2} at the distance l divided by the number of such pairs. Here br(ρ_{1}, ρ_{2}) stands for the number of vertices used by both 2breaks ρ_{1} and ρ_{2}.
Since the rearrangements on branches of the phylogenetic tree are unknown, we use the following sampling procedure to approximate R(l). Given genomes P and Q, we sample various shortest rearrangement scenarios between these genomes by generating random 2break transformations of P into Q. To generate a random transformation we first randomly select a nontrivial cycle C in the breakpoint graph G(P, Q) with the probability proportional to C/ = 2  1, that is, the number of 2breaks required to transform such a cycle into a collection of trivial cycles (C stands for the length of C). Then we uniformly randomly select a 2break ρ from the set of all ${(}_{2}^{{}^{\text{C}/2}})=\frac{\text{C}(\text{C}2)}{8}$ 2breaks that splits the selected cycle C into 2 8 two and thus by Theorem 1 decreases the distance between P and Q by one (that is, d(ρ P, Q) = d(P, Q) 1). We continue selecting nontrivial cycles and 2breaks in an iterative fashion for genomes ρ · P and Q and so on until P is transformed into Q.
The described sampling can be performed for every branch e = (P, Q) of the phylogenetic tree, essentially partitioning e into length(e) = d(P, Q) subbranches, each featuring a single 2break. The resulting tree will have ∑_{ e }length(e) subbranches, where the sum is taken over all branches e.
For each pair of subbranches, we compute the number of reused vertices across them and accumulate these numbers according to the distance between these subbranches in the tree. The empirical multispecies breakpoint reuse (the average reuse between all subbranches at the distance l) is defined as the actual multispecies breakpoint reuse in a sampled rearrangement scenario. Figure S2 in Additional file 1 represents this function for five simulated genomes on m = 2, 000 synteny blocks, n = 900 fragile regions, and the turnover rate x varying from zero to four, with the same phylogenetic tree and distances between the genomes (averaged over 100 random samplings, while individual samplings produce varying results, we found that the variance of the R(l) estimates across various samplings is rather small). Figure S3 in Additional file 1 demonstrates that our sampling procedure, while imperfect, accurately estimates the theoretical R(l) curve (see [52] for other approaches to sampling rearrangement scenarios). Similar tests on phylogenetic trees with varying topologies demonstrated a good fit between actual, empirical, and theoretical R(l) curves (data are not shown).
We argue that the empirical multispecies breakpoint reuse curve R(l) complements the 'exponential length distribution' [2] and 'pairwise breakpoint reuse' [3] tests as the third criterion to accept/reject RBM, FBM, and now TFBM. One can use the parameters n and x (estimated from empirical R(l) curve) to evaluate the extent of the 'birth and death' process and to explain why Ma et al. [33] found so few shared breakpoints between different mammalian lineages. In practice, the 'multispecies breakpoint reuse test' can be applied in the same way as the NadeauTaylor 'exponential length distribution test' was applied in numerous papers. The NadeauTaylor test typically amounted to constructing a histogram of synteny blocks and evaluating (often visually) whether it fits the exponential distribution. Similarly, the 'multispecies breakpoint reuse test' amounts to constructing R(l) curve and evaluating whether it significantly deviates from a horizontal line suggested by RBM and FBM. The estimated parameters of the TFBM model (see Methods) can be used to quantify the extent of these deviations.
TFBM also raises an intriguing question of what triggers the birth and death of fragile regions. As demonstrated by Zhao and Bourque [38], the disproportionately large number of rearrangements in primate lineages are flanked by MSDs. TFBM is consistent with the ZhaoBourque hypothesis that rearrangements are triggered by MSDs since MSDs are also subject to the 'birth and death' process. Indeed, after a segmental duplication the pair of matching segments becomes subjected to random mutations and the similarity between these segments dissolves with time (a pair of segmental duplications 'disappears' after approximately 40 million years of evolution if one adopts the parameters for defining segmental duplications from [53]).
The mosaic structure of segmental duplications [53] provides an additional explanation of how MSDs may promote breakpoint reuses and generate long cycles typical for the breakpoint graphs of mammalian genomes. The future studies of the correlation between fragile regions and MSDs in the human genome will benefit from the algorithms for precise detection of rearrangement breakpoints [54] and will be described elsewhere.
Fragile regions in the human genome
Imagine the following gedanken experiment: 25 million years ago (time of the humanmacaque split) a scientist sequences the genome of the humanmacaque ancestor (QH) and attempts to predict the sites of (future) rearrangements in the (future) human genome. The only other information the scientist has is the mouse, rat, and dog genomes. While RBM offers no clues on how to make such a prediction, FBM suggests that the scientist should use the breakpoints between one of the available genomes and QH as a proxy for fragile regions. For example, there are 552 breakpoints between the mouse genome (M) and QH and 34 of them were actually used in the human lineage, resulting in only 34 = 552 ≈ 6% accuracy in predicting future human breakpoints (we use synteny blocks larger than 500 K from [50]).
TFBM suggests that the scientist should rather use the closest genome to QH to better predict the human breakpoints. That can be achieved by first reconstructing the common ancestor (MRD) of mouse, rat, dog, and humanmacaque ancestor and then using the breakpoints between MRD and QH as a proxy for the sites of rearrangements in the human lineage. 18 out 162 breakpoints between MRD and QH were used in the human lineage, resulting in 18 = 162 ≈ 11% accurate prediction of human breakpoints, nearly doubling the accuracy of predictions from distant genomes.
Now imagine that the scientist somehow gained access to the extant macaque genome. There are 68 breakpoints between Q and QH and 10 of them were used in the human lineage, resulting in 10 = 68 ≈ 16% accurate prediction of human breakpoints, again improving the accuracy of predictions. These estimates indicate that TFBM can be used to improve the prediction accuracy of future rearrangements in various lineages and demonstrate that the sites of recent rearrangements in the human and other primate lineages represent the best guess for the currently active fragile regions in the human genome.
Conclusions
Since every species on Earth (including Homo sapiens) may speciate into multiple new species, one can ask a question: 'How will the human genome evolve in the next million years?' TFBM suggests the putative sites of future rearrangements in the human genome. The answer to the question 'Where are the (future) fragile regions in the human genome?' may be surprisingly simple: they are likely to be among the breakpoint regions that were used in various primate lineages.
Nadeau and Taylor [2] proposed RBM based on a single observation: the exponential distribution of the humanmouse synteny block sizes. There is no doubt that jumping to this conclusion was not fully justified: there are many other models (for example, FBM) that lead to the same exponential distribution of the 'visible' synteny block sizes. Currently, there is no single piece of evidence that would allow one to claim that RBM is correct and FBM is not.
While Pevzner and Tesler [3] revealed large breakpoint reuse (supporting FBM and contradicting RBM), Ma et al. [33] revealed low breakpoint interreuse (contradicting FBM). This discovery calls for yet another generalization of FBM. The proposed TFBM model not only passes both 'exponential length distribution' test (motivation for RBM) and 'pairwise breakpoint reuse' test (motivation for FBM) but also explains the puzzling discovery of limited breakpoint interreuse in [33]. We therefore argue that TFBM is a more accurate model of chromosome evolution, allowing one to approximate the currently active fragile regions in the human genome.
Needless to say, TFBM, similarly to RBM and FBM (or various models of point mutations, for example, JukesCantor model), is a simplistic model of chromosome evolution that is only an approximation of the real evolutionary process. Moreover, in the current paper we considered TFBM only for the case of 2breaks and did not include other rearrangements such as transpositions. However, it is fair to assume that transpositions are as likely to happen on incident branches as on distant branches, implying that they cannot possibly cause the reduced breakpoint interreuse on distant branches. In addition to limitations of TFBM as a model, there exists a concern whether computation of empirical multispecies breakpoint reuse (that requires reconstruction of ancestral genomes) may be affected by errors in reconstruction of ancestral genomes. While various tools for ancestral genome reconstruction (such as MGRA [50] and inferCARs [33]) were shown to be quite accurate (in particular, they produce nearly identical results while using very different algorithms), it is a challenging open problem to evaluate the multispecies breakpoint reuse without explicitly computing ancestral genomes.
The key point of this paper is the birth and death process of fragile regions rather than a specific model aimed at estimating the hidden parameters of this process. TFBM is merely an initial and oversimplistic attempt to estimate these parameters. The parameters predicted by TFBM (for example, the number of active fragile regions) are currently difficult to superimpose with scarce information about rearrangements in only seven reliably completed mammalian genomes, not unlike the parameters of RBM derived in 1984 when no highresolution comparative mammalian genomic architectures were available. However, similarly to comparative mapping efforts in early 1990 s that confirmed the NadeauTaylor estimates, we believe that imminent sequencing of over 400 primate species will soon provide the detailed information about chromosomal fragility in human genome and will allow one to verify the TFBM parameters.
Similarly to the discovery of breakpoint reuse in 2003 [3], there is currently only indirect evidence supporting the birth and death of fragile regions in chromosome evolution. However, we hope that, similarly to FBM (that led to many followup studies supporting the existence of fragile regions), TFBM will trigger further investigations of the fragile regions longevity.
Materials and methods
Computing multispecies breakpoint reuse in the TFBM model
Let Fragile and Solid be the sets of n initial fragile regions and m  n initial solid regions respectively. In TFBM, the sets Fragile and Solid change in accordance with the turnover rate x, that is, after every 2break x fragile regions (corresponding to 2x vertices in the breakpoint graph) from Fragile are moved to Solid and vice versa.
Figure S6 in Additional file 1 demonstrates that this formula fits simulated data well, thus opening a possibility to determine the parameters m, n, and x for given real genomes.
We remark that if $\frac{xml}{n(mn)}\ll 1$ is approximated by a line $\frac{8\cdot (mn)}{n\cdot m}\left(1\frac{xm}{n(mn)}\ell \right)+\frac{8}{m}=\frac{8}{n}\frac{8x}{{n}^{2}}\ell $ that does not depend on m.
The difference between empirical and theoretical estimates for R(l)
Figure S3 in Additional file 1 illustrates the results of simulating of 400 2breaks according to TFBM with parameters m = 2, 000, n = 900, x = 1. As expected, the theoretical curve and the curve derived from simulated data (without sampling of various rearrangement scenarios) are nearly identical. We now assume that only five out of 401 simulated genomes are available (after 0, 100, 200, 300, and 400 rearrangements) and use sampling of rearrangement scenarios to compute the empirical R(l) (Figure S3 in Additional file 1). One can see that empirical R(l) differs from the theoretical R(l), particularly for small'. To understand why the empirical curve (obtained via sampling of rearrangement scenarios) differs from the theoretical curve, one has to realize that the multispecies breakpoint reuse test requires multiple genome to reveal the 'birth and death' of fragile regions. Indeed, it is impossible to detect this process from only two genomes: for example, sampling of rearrangement scenarios on a single branch (simulated with TFBM with parameters described above) produces a nearly horizontal curve R(l) ≈ 0.0083 with TFBM signal lost. The green curve follows the same horizontal trend for small l (for example l < 100) that typically represent pairs of 2breaks on the same branch. However, for distances larger than the shortest branches, the theoretical curve approximates the empirical R(l) curve well. The reason this 'horizontal trend' is not seen in Figure 11 most likely explained by the fact that H+ and Q+ branches in the corresponding phylogenetic tree are rather short thus masking this effect.
Abbreviations
 FBM:

fragile breakage model
 MSDs:

matching segmental duplications
 RBM:

random breakage model
 TFBM:

turnover fragile breakage model.
Declarations
Acknowledgements
The authors thank Glenn Tesler and Jian Ma for many helpful comments.
Authors’ Affiliations
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