Modeling the evolution space of breakage fusion bridge cycles with a stochastic folding process

Breakage-Fusion-Bridge cycles in cancer arise when a broken segment of DNA is duplicated and an end from each copy joined together. This structure then 'unfolds' into a new piece of palindromic DNA. This is one mechanism responsible for the localised amplicons observed in cancer genome data. The process has parallels with paper folding sequences that arise when a piece of paper is folded several times and then unfolded. Here we adapt such methods to study the breakage-fusion-bridge structures in detail. We firstly consider discrete representations of this space with 2-d trees to demonstrate that there are 2^(n(n-1)/2) qualitatively distinct evolutions involving n breakage-fusion-bridge cycles. Secondly we consider the stochastic nature of the fold positions, to determine evolution likelihoods, and also describe how amplicons become localised. Finally we highlight these methods by inferring the evolution of breakage-fusion-bridge cycles with data from primary tissue cancer samples

A sequence-based survey of the complex structural organization of tumor genomes

Tumors and cancer cell lines were surveyed with end-sequencing profiling, yielding the largest available collection of sequence-ready tumor genome breakpoints and providing evidence that some rearrangements may be recurrent

The Combinatorics of Tandem Duplication

Tandem duplication is an evolutionary process whereby a segment of DNA is replicated and proximally inserted. The different configurations that can arise from this process give rise to some interesting combinatorial questions. Firstly, we introduce an algebraic formalism to represent this process as a word producing automaton. The number of words arising from n tandem duplications can then be recursively derived. Secondly, each single word accounts for multiple evolutions. With the aid of a bi-coloured 2d- tree, a Hasse diagram corresponding to a partially ordered set is constructed, from which we can count the number of evolutions corresponding to a given word. Thirdly, we implement some subtree prune and graft operations on this structure to show that the total number of possible evolutions arising from n tandem duplications is $\prod_{k=1}^n(4^k - (2k + 1))$. The space of structures arising from tandem duplication thus grows at a super-exponential rate with leading order term $\mathcal{O}(4^{\frac{1}{2}n^2})$

Genome reconstruction and combinatoric analyses of rearrangement evolution

Cancer is often associated with a high number of large-scale, structural rearrangements. In a highly selective environment, some `driver' mutations conferring clonal growth advantage will be positively selected, accounting for further cancer development. Clarifying their nature, as well as their contribution to the pathology is a major current focus of biomedical research. Next generation sequencing technologies can be used nowadays to generate high-resolution data-sets of these alterations in cancer genomes. This project has been developed along two main lines: 1) the reconstruction of cancer aberrant karyotypes, together with their underlying evolutionary history; 2) the elucidation of some combinatorial properties associated with gene duplications. We applied graph theory to the problem of reconstructing the final cancer genome sequence; additionally, we developed an algorithmic approach for the reconstruction of a multi-step evolution consistent with read coverage and paired end data, giving insights on the possible molecular mechanisms underlying rearrangements. Looking at the combinatorics of both tandem and inverted duplication, we developed an algebraic formalism for the representation of these processes. This allowed us to both explore the geometric properties of sequences arising by Tandem Duplication (TD), and obtain a recursion for the number of tandem duplications evolutions after n events. Such results are missing for inverted duplications, whose combinatorial properties have been nevertheless deeply elucidated. Our results have allowed: 1) the identification, through an original approach, of potential rearrangement mechanisms associated with cancer development, and 2) the definition and mathematical description of the complete evolutionary space of specific rearrangement classes