A symmetry-inclusive algebraic approach to genome rearrangement

Terauds, Venta; Stevenson, Joshua; Sumner, Jeremy G.

doi:10.1142/s0219720021400151

Cited by 5 publications

(13 citation statements)

References 25 publications

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“…The genome algebra framework was presented in Terauds and Sumner [30], along with details of the construction for the case of unsigned circular genomes with dihedral symmetry. Here we consider the algebra for circular single-chromosome genomes with oriented regions, both with and without an origin of replication.…”

Section: Genome Instances and Permutation Cloudsmentioning

confidence: 99%

“…We define the path probabilities for a genome zσ i with respect to the reference genome z via α k (zσ i ) := probability of obtaining zσ i from z via k rearrangement events . Now, these can be read from the first column of the matrix ρ( s) k , or obtained via the trace of a modified matrix [30]:…”

Section: Genome Instances and Permutation Cloudsmentioning

confidence: 99%

“…) . To compute this more efficiently, one would usually, as a first step, decompose the algebra into its irreducible modules; see Terauds and Sumner [30] for details.…”

Section: Genome Instances and Permutation Cloudsmentioning

confidence: 99%

“…The only other condition that we put on the model is model reversibility, that is, that whenever zaz ∈ M, one must have za −1 z ∈ M with w(zaz) = w(za −1 z). This is not required for any of the above, however it ensures that the Markov process is reversible [30], so that σ −1 can be replaced by σ in (1). Further, it allows the path probabilities to be more efficiently computed via diagonalising the respective matrices.…”

Section: Genome Instances and Permutation Cloudsmentioning

confidence: 99%

“…Rearrangement models used in practice have tended to have this property as a consequence of being quite general, for example models consisting of all inversions, all inversions of size k, all inversions with some probability p and all transpositions with probability q, and so on. However, until recently, [29,30] this has not been included in the theoretical framework.…”

Section: Introductionmentioning

confidence: 99%

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A symmetry-inclusive algebraic approach to genome rearrangement

Terauds¹,

Stevenson²,

Sumner³

2021

Preprint

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Of the many modern approaches to calculating evolutionary distance via models of genome rearrangement, most are tied to a particular set of genomic modelling assumptions and to a restricted class of allowed rearrangements. The "position paradigm", in which genomes are represented as permutations signifying the position (and orientation) of each region, enables a refined model-based approach, where one can select biologically plausible rearrangements and assign to them relative probabilities/costs. Here, one must further incorporate any underlying structural symmetry of the genomes into the calculations and ensure that this symmetry is reflected in the model. In our recently-introduced framework of genome algebras, each genome corresponds to an element that simultaneously incorporates all of its inherent physical symmetries. The representation theory of these algebras then provides a natural model of evolution via rearrangement as a Markov chain. In this paper, we consider the relevant genome algebras for signed circular single-chromosome genomes with and without an origin of replication, and demonstrate the computation of genomic distance, as estimated via several distance measures, under various rearrangement models.

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Section: Genome Instances and Permutation Cloudsmentioning

confidence: 99%

Section: Genome Instances and Permutation Cloudsmentioning

confidence: 99%

“…) . To compute this more efficiently, one would usually, as a first step, decompose the algebra into its irreducible modules; see Terauds and Sumner [30] for details.…”

Section: Genome Instances and Permutation Cloudsmentioning

confidence: 99%

Section: Genome Instances and Permutation Cloudsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A symmetry-inclusive algebraic approach to genome rearrangement

Terauds¹,

Stevenson²,

Sumner³

2021

Preprint

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A new algebraic approach to genome rearrangement models

Terauds

Sumner

2022

J. Math. Biol.

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We present a unified framework for modelling genomes and their rearrangements in a genome algebra, as elements that simultaneously incorporate all physical symmetries. Building on previous work utilising the group algebra of the symmetric group, we explicitly construct the genome algebra for the case of unsigned circular genomes with dihedral symmetry and show that the maximum likelihood estimate (MLE) of genome rearrangement distance can be validly and more efficiently performed in this setting. We then construct the genome algebra for a more general case, that is, for genomes that may be represented by elements of an arbitrary group and symmetry group, and show that the MLE computations can be performed entirely within this framework. There is no prescribed model in this framework; that is, it allows any choice of rearrangements that preserve the set of regions, along with arbitrary weights. Further, since the likelihood function is built from path probabilities—a generalisation of path counts—the framework may be utilised for any distance measure that is based on path probabilities.

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Rearrangement Events on Circular Genomes

Stevenson,

Terauds,

Sumner

2023

Bull Math Biol

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Early literature on genome rearrangement modelling views the problem of computing evolutionary distances as an inherently combinatorial one. In particular, attention is given to estimating distances using the minimum number of events required to transform one genome into another. In hindsight, this approach is analogous to early methods for inferring phylogenetic trees from DNA sequences such as maximum parsimony—both are motivated by the principle that the true distance minimises evolutionary change, and both are effective if this principle is a true reflection of reality. Recent literature considers genome rearrangement under statistical models, continuing this parallel with DNA-based methods, with the goal of using model-based methods (for example maximum likelihood techniques) to compute distance estimates that incorporate the large number of rearrangement paths that can transform one genome into another. Crucially, this approach requires one to decide upon a set of feasible rearrangement events and, in this paper, we focus on characterising well-motivated models for signed, uni-chromosomal circular genomes, where the number of regions remains fixed. Since rearrangements are often mathematically described using permutations, we isolate the sets of permutations representing rearrangements that are biologically reasonable in this context, for example inversions and transpositions. We provide precise mathematical expressions for these rearrangements, and then describe them in terms of the set of cuts made in the genome when they are applied. We directly compare cuts to breakpoints, and use this concept to count the distinct rearrangement actions which apply a given number of cuts. Finally, we provide some examples of rearrangement models, and include a discussion of some questions that arise when defining plausible models.

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A symmetry-inclusive algebraic approach to genome rearrangement

Cited by 5 publications

References 25 publications

A symmetry-inclusive algebraic approach to genome rearrangement

A symmetry-inclusive algebraic approach to genome rearrangement

A new algebraic approach to genome rearrangement models

Rearrangement Events on Circular Genomes

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