^{ * }

MAA and PAP conceived and designed the experiments, analyzed the data, and wrote the paper. MAA performed the experiments.

The authors have declared that no competing interests exist.

In a landmark paper, Nadeau and Taylor [

In 1970, Susumu Ohno came up with two fundamental models of chromosome evolution that were subject to many controversies in the last 35 years [

Rearrangements are genomic “earthquakes” that change the chromosomal architectures. The fundamental question in molecular evolution is whether there exist “chromosomal faults” (rearrangement hotspots) where rearrangements are happening over and over again. RBM postulates that rearrangements are “random,” and thus there are no rearrangement hotspots in mammalian genomes.

For the sake of completeness, we give a simple version of both the Pevzner-Tesler and Sankoff-Trinh arguments. Shortly after the human and mouse genomes were sequenced, Pevzner and Tesler [

These results are in conflict with the classical Nadeau and Taylor [

Sankoff and Trinh [

Recently, Peng et al. [

The standard rearrangement operations (i.e., reversals, translocations, fusions, fissions) can be modelled by making two breaks in a genome and gluing the resulting fragments in a new order. One can imagine a hypothetical

We recently proved the duality theorem for the

We start our analysis with _{1},...,x_{n}_{i}_{i}^{t}_{i}^{h}_{i}^{t}_{i}^{h}

The breakpoint graph

Let

A 2-break on edges (

Let ^{t},x^{h}

The _{k}

Different from the genomic distance problem [

_{2}_{2}_{2}

While 2-breaks correspond to standard rearrangements, 3-breaks add transposition-like operations (transpositions and inverted transpositions) as well as three-way fissions to the set of rearrangements (

A 3-break on edges (

A transposition cuts off a segment of one chromosome and inserts it into the same or another chromosome. A transposition of a segment _{i}π_{i+1}…π_{j}_{1}…π_{i-1}_{i}π_{i+1}…π_{j}_{j+1}…π_{k-1}π_{k}…π_{n}_{1}…π_{i-1}π_{j+1}…π_{k-1}_{i}π_{i+1}…π_{j}_{κ}…π_{n}_{1}…π_{m}_{1}…σ_{n}_{i}π_{i+1}…π_{j}_{1}…π_{i-1}π_{j+1}π_{j+2}…π_{m}_{1}…σ_{k-1}_{i}π_{i+1}…π_{j}_{k}…σ._{n}

Let ^{odd}

^{odd}

^{odd}_{3}^{odd}_{3}^{odd}_{3}^{odd}

For the sake of completeness, below we formulate the duality theorem for the _{k}_{3}^{odd}

Sankoff summarized arguments against FBM in the following sentence [

“...And we cannot infer whether mutually randomized synteny block orderings derived from two divergent genomes were created through bona fide breakpoint re-use or rather through noise introduced in block construction or through processes other than reversals and translocations.”

Below we consider the “other processes” argument. The “noise in block construction” argument consists of two parts: synteny block generation and gene deletion. The flaw in the first argument was revealed in [

In this paper, we study transformations between the human genome

The breakpoint graph _{2}_{3}

The rebuttal of RBM raises a question about finding a transformation of

_{2}(P,Q) breaks. Moreover, there exists a series of d_{3}(P,Q) 3-breaks transforming P into Q that makes d_{3}(P,Q) + d_{2}(P,Q) breaks.

_{2}

Consider a shortest series of complete 3-breaks transforming every odd black-gray cycle into a trivial cycle and every even black-gray cycle into trivial cycles and a single cycle with two black edges. This series consists of _{3}^{even}^{even}^{even}_{3}_{3}^{even}^{even}_{3}^{even}_{3}_{2}

_{3}_{2}

Theorem 4 implies that any transformation of the human genome _{2}_{2}_{3}_{2}

_{2}(P,Q)_{3}(P,Q)}. Moreover, there exists a series of max{d_{2}(P,Q)_{3}(P,Q)} 3-breaks transforming P into Q with at most t complete 3-breaks.

_{2}_{2}

Consider a series of complete 3-breaks, transforming every black-gray cycle with _{3}^{even}^{even}^{even}_{3}^{even}_{2}_{2}_{2}_{3}^{even}_{2}_{3}

Theorems 4 and 6 imply:

_{2}(P,Q)_{2}(P,Q)_{3}(P,Q)} breaks. In particular, any such series of 3-breaks with t ≤ d_{2}(P,Q)_{3}(P,Q) complete 3-breaks makes at least 2d_{2}(P,Q)

Corollary 7 gives the lower bound for the breakpoint re-use rate as a function of the number of complete 3-breaks (i.e., transpositions and three-way fissions) in a series of 3-breaks transforming one genome into the other. For the human genome

A lower bound for the breakpoint re-use rate as a function of the number of complete 3-breaks in a series of 3-breaks between the circularized human and mouse genomes based on 281 conserved segments from [

In the case of linear genomes, the plot is similar, with the breakpoint re-use rate of ≈0.1 lower than in the circular case [

Corollaries 5 and 7 address only the case of circularized chromosomes and further analysis is needed to extend it to the case of linear chromosomes (see [

The papers [

“...In fact, Pavel Pevzner (personal communication) has pointed out likely errors in our simulation procedure. Subsequent experiments showed that with realistic sizes and numbers of short inversions, unrealistically large number of long inversions were necessary for the amalgamation process to have an effect...”

Despite the importance of choosing realistic parameters, the paper [

Sankoff and Trinh [

(A) Breakpoint re-use rate for parameters

(B) Breakpoint re-use rate for parameters

The inability to connect

(A) Synteny block sizes (for a permutation with 1,000 elements after 320 reversals) do not fit the exponential distribution expected from RBM.

(B) Synteny block sizes (for a permutation with 25,000 elements after 320 reversals) fit the exponential distribution expected from RBM.

This particular deficiency of the Sankoff-Trinh simulations is easy to fix: one should simply increase the granularity (i.e., increase

This problem did not escape the attention of Pevzner and Tesler [

(A) Breakpoint re-use rate as a function of the maximal size of deleted synteny blocks (as the proportion of the whole genome length). Deletion of blocks shorter than 1 Mb as in [

(B) The distribution of breakpoint re-use at

We admit that since the choice of 1 Mb (

Indeed, in this case all synteny blocks shorter than 5 Mb would have to be deleted, and thus would have to be declared to be the breakpoint regions rather than the synteny blocks (for

We emphasize that ^{i}

To better compare the Sankoff-Trinh gene deletion process with the synteny block deletion process, one may switch to parameter

Nadeau and Taylor [

Perusal of the UCSC Genome Browser (

When RBM was formalized in 1984 [

Distribution of the synteny block sizes between the human and mouse genomes based on (A) 281 synteny blocks from [

The paper [

If RBM is put to rest in favor of FBM, one has to answer the question of what makes certain regions break and others not break. Peng et al. [

(123 KB PDF)

We are indebted to Glenn Tesler, who kindly provided us with a detailed review of roughly the same length as this paper. We are also grateful to Vikas Bansal, Tzvika Hartman, and Alex Zelikovsky for insightful comments. We are indebted to David Sankoff for insightful critical arguments in [

fragile breakage model

random breakage model