Reducibility of Gene Patterns in Ciliates using the Breakpoint Graph

Gene assembly in ciliates is one of the most involved DNA processings going on in any organism. This process transforms one nucleus (the micronucleus) into another functionally different nucleus (the macronucleus). We continue the development of the theoretical models of gene assembly, and in particular we demonstrate the use of the concept of the breakpoint graph, known from another branch of DNA transformation research. More specifically: (1) we characterize the intermediate gene patterns that can occur during the transformation of a given micronuclear gene pattern to its macronuclear form; (2) we determine the number of applications of the loop recombination operation (the most basic of the three molecular operations that accomplish gene assembly) needed in this transformation; (3) we generalize previous results (and give elegant alternatives for some proofs) concerning characterizations of the micronuclear gene patterns that can be assembled using a specific subset of the three molecular operations.Comment: 30 pages, 13 figure

The Fibers and Range of Reduction Graphs in Ciliates

The biological process of gene assembly has been modeled based on three types of string rewriting rules, called string pointer rules, defined on so-called legal strings. It has been shown that reduction graphs, graphs that are based on the notion of breakpoint graph in the theory of sorting by reversal, for legal strings provide valuable insights into the gene assembly process. We characterize which legal strings obtain the same reduction graph (up to isomorphism), and moreover we characterize which graphs are (isomorphic to) reduction graphs.Comment: 24 pages, 13 figure

Strategies of Loop Recombination in Ciliates

Gene assembly in ciliates is an extremely involved DNA transformation process, which transforms a nucleus, the micronucleus, to another functionally different nucleus, the macronucleus. In this paper we characterize which loop recombination operations (one of the three types of molecular operations that accomplish gene assembly) can possibly be applied in the transformation of a given gene from its micronuclear form to its macronuclear form. We also characterize in which order these loop recombination operations are applicable. This is done in the abstract and more general setting of so-called legal strings.Comment: 22 pages, 14 figure

The Matrix Sortability Problem

Sorting is such a fundamental component of achieving efficiency that a significant body of mathematics is dedicated to the investigation of sorting. Any modern textbook on algorithms contains chapters on sorting. One approach to arranging a disorganized list of items into an organized list is to successively identify two blocks of contiguous items, and swap the two blocks. In a fundamental paper D.A. Christie showed that a special version of block swapping, in recent times called context directed swapping and abbreviated cds, is the most efficient among block swapping strategies to achieve an organized list of items. The cds sorting strategy is also the most robust among block swap based sorting methods. It has been discovered that the context directed block swap operation on a list of objects generalizes to an operation on simple graphs. In turn it has been discovered that this operation on simple graphs corresponds with an operation on the adjacency matrix of a simple graph. The adjacency matrix is a symmetric square matrix with entries 0 and 1, and all diagonal entries 0. The corresponding operation is denoted Mcds, abbreviating matrix context directed swap. The operation on the adjacency matrix naturally employs the arithmetic of GF(2), the finite field of two elements. It has been speculated that the Mcds operation on these specific matrices over GF(2) corresponds with the more than a century old Schur complement operation on these matrices. In this thesis, we confirm this prior speculation about the correspondence between Mcds and the Schur complement, in the context of GF(2). We generalize the Mcds operation to not necessarily square matrices over arbitrary fields and we prove that the generalized Mcds corresponds with the Schur complement also in the more general context of all fields

Doctor of Philosophy

dissertationGenotype Phenotype Association (GPA) is a means to identify candidate genes and genetic variants that may contribute to phenotypic variation. Technological advances in DNA sequencing continue to improve the efficiency and accuracy of GPA. Currently, High Throughput Sequencing (HTS) is the preferred method for GPA as it is fast and economical. HTS allows for population-level characterization of genetic variation, required for GPA studies. Despite the potential power of using HTS in GPA studies, there are technical hurdles that must be overcome. For instance, the excessive error rate in HTS data and the sheer size of population-level data can hinder GPA studies. To overcome these challenges, I have written two software programs for the purpose of HTS GPA. The first toolkit, GPAT++, is designed to detect GPA using small genetic variants. Unlike pervious software, GPAT++'s association test models the inherent errors in HTS, preventing many spurious GPA. The second toolkit, Whole Genome Alignment Metrics (WHAM), was designed for GPA using large genetic variants (structural variants). By integrating both structural variant identification and association testing, WHAM can identify shared structural variants associated with a phenotype. Both GPAT++ and WHAM have been successfully applied to real-world GPA studie

Models of natural computation : gene assembly and membrane systems

This thesis is concerned with two research areas in natural computing: the computational nature of gene assembly and membrane computing. Gene assembly is a process occurring in unicellular organisms called ciliates. During this process genes are transformed through cut-and-paste operations. We study this process from a theoretical point of view. More specifically, we relate the theory of gene assembly to sorting by reversal, which is another well-known theory of DNA transformation. In this way we obtain a novel graph-theoretical representation that provides new insights into the nature of gene assembly. Membrane computing is a computational model inspired by the functioning of membranes in cells. Membrane systems compute in a parallel fashion by moving objects, through membranes, between compartments. We study the computational power of various classes of membrane systems, and also relate them to other well-known models of computation.Netherlands Organisation for Scientific Research (NWO), Institute for Programming research and Algorithmics (IPA)UBL - phd migration 201

Sorting Permutations: Games, Genomes, and Cycles

Permutation sorting, one of the fundamental steps in pre-processing data for the efficient application of other algorithms, has a long history in mathematical research literature and has numerous applications. Two special-purpose sorting operations are considered in this paper: context directed swap, abbreviated cds, and context directed reversal, abbreviated cdr. These are special cases of sorting operations that were studied in prior work on permutation sorting. Moreover, cds and cdr have been postulated to model molecular sorting events that occur in the genome maintenance program of certain species of single-celled organisms called ciliates. This paper investigates mathematical aspects of these two sorting operations. The main result of this paper is a generalization of previously discovered characterizations of cds-sortability of a permutation. The combinatorial structure underlying this generalization suggests natural combinatorial two-player games. These games are the main mathematical innovation of this paper.Comment: to appear in Discrete Mathematics, Algorithms and Application