Linearization of ancestral multichromosomal genomes

Abstract

Background Recovering the structure of ancestral genomes can be formalized in terms of properties of binary matrices such as the Consecutive-Ones Property (C1P). The Linearization Problem asks to extract, from a given binary matrix, a maximum weight subset of rows that satisfies such a property. This problem is in general intractable, and in particular if the ancestral genome is expected to contain only linear chromosomes or a unique circular chromosome. In the present work, we consider a relaxation of this problem, which allows ancestral genomes that can contain several chromosomes, each either linear or circular. Result We show that, when restricted to binary matrices of degree two, which correspond to adjacencies, the genomic characters used in most ancestral genome reconstruction methods, this relaxed version of the Linearization Problem is polynomially solvable using a reduction to a matching problem. This result holds in the more general case where columns have bounded multiplicity, which models possibly duplicated ancestral genes. We also prove that for matrices with rows of degrees 2 and 3, without multiplicity and without weights on the rows, the problem is NP-complete, thus tracing sharp tractability boundaries. Conclusion As it happened for the breakpoint median problem, also used in ancestral genome reconstruction, relaxing the definition of a genome turns an intractable problem into a tractable one. The relaxation is adapted to some biological contexts, such as bacterial genomes with several replicons, possibly partially assembled. Algorithms can also be used as heuristics for hard variants. More generally, this work opens a way to better understand linearization results for ancestral genome structure inference.

Ancestral gene synteny reconstruction improves extant species scaffolding

“…This overview is lost if we force the data to fit in a linear structure. But if a linear ancestral genome is really needed, linearization techniques exist [37], even if we would argue for linearization techniques that also put into question the input data.…”

Assessing the Robustness of Parsimonious Predictions for Gene Neighborhoods from Reconciled Phylogenies: Supplementary Material

“…This is consistent with the fact that for rare evolutionary events such as genome rearrangements, a parsimony approach is relevant, especially when it can be complemented by efficient algorithms to explore slightly sub-optimal solutions, such as DeClone, and to explore the parameter space. In terms of direct applications of the method developed here and in [6], gene-tree based reconstruction of ancestral gene orders comes to mind [5]; more precisely, ancestral adjacencies could be determined and scored using a mixture of their Boltzmann probability (that can be computed efficiently using DeClone) and robustness to changes of the cost scheme, and conflicts could be cleared out independently and efficiently for each ancestral species using the algorithm of [12] for example. An interesting observation is that even the set of ancestral adjacencies that are universally-parsimonious and robust to changes in the cost scheme contains a significant number of adjacencies participating in syntenic conflict.…”

The SCJ Small Parsimony Problem for Weighted Gene Adjacencies

“…If α = 1, i.e., we do not take evolution in terms of SCJ distance along the branches of the tree into account, we can solve the problem by applying independently a maximum-weight matching algorithm at each internal node [26]. So the extreme cases of the problem are tractable, and while we assume that the general problem is hard, we will now prove it for a small range of α. Theorem 1.…”