Niklas Eriksen

(1 + ɛ)-Approximation of sorting by reversals and transpositions

Gu et al. gave a 2-approximation for computing the minimal number of inversions and transpositions needed to sort a permutation. There is evidence that, from the point of view of computational molecular biology, a more adequate objective function is obtained, if transpositions are given double weight. We present a (1 + e)-approximation for this problem, based on the exact algorithm of Hannenhalli and Pevzner for sorting by reversals only.

Measuring genome divergence in bacteria: A case study using Chlamydian data

We have studied the relative contribution of inversions, transpositions, deletions, and nucleotide substitutions to the evolution of Chlamydia trachomatis and Chlamydia pneumoniae. The minimal number of rearrangement events required for converting the gene order structure of one genome into that of the other was estimated to 59 +/- 6 events, including 13% inversions, 38% short inversions, and 49% transpositions. In contrast to previous findings, no examples of horizontal gene transfer subsequent to species divergence were identified, nor any evidence for an excessive number of tandem gene duplications. A statistical model was used to compare nucleotide frequencies for a set of genes uniquely present in one species to a set of orthologous genes present in both species. The two data sets were not significantly different, which is indicative of a low frequency of horizontal gene transfer events. This is based on the assumption that a foreign gene of different nucleotide content will not have become completely ameliorated, as verified by simulations of the amelioration rate at twofold and fourfold degenerate codon sites. The frequencies of nucleotide substitutions at twofold and fourfold degenerate sites, deletions, inversions, and translocations were estimated to 1.42, 0.62, 0.18, 0.01, and 0.01 per site, respectively.

Reversal and transposition medians

In determining phylogenetic trees using gene order information, medians provide a powerful alternative to pairwise distances. On the other hand, both breakpoint and reversal medians are NP-hard to compute and the use of medians has been limited to relatively closely related genomes. In this paper, we show that in spite of the greater non-uniqueness of reversal medians, compared to breakpoint medians, medians of moderately distant genomes are often widely spread. This means that regardless of which approximation algorithms one may devise for computing reversal medians, the genomes need to be closely related for phylogenetic tree computations to be successful. To show this, we use results on transposition medians, which behave similarly, and also support our claims with simulations and a real data example with widely spread medians.

(1+epsilon)-Approximation of Sorting by Reversals and Transpositions

Gu et al. gave a 2-approximation for computing the minimal number of inversions and transpositions needed to sort a permutation. There is evidence that, from the point of view of computational molecular biology, a more adequate objective function is obtained, if transpositions are given double weight. We present a (1 + Ɛ)-approximation for this problem, based on the exact algorithm of Hannenhalli and Pevzner, for sorting by reversals only.

Approximating the Expected Number of Inversions Given the Number of Breakpoints

We look at a problem with motivation from computational biology: Given the number of breakpoints in a permutation (representing a gene sequence), compute the expected number of inversions that have occurred. For this problem, we obtain an analytic approximation that is correct within a percent or two. For the inverse problem, computing the expected number of breakpoints after any number of inversions, we obtain an analytic approximation with an error of less than a hundredth of a breakpoint.

Expected gene-order distances and model selection in bacteria

MOTIVATION The evolutionary distance inferred from gene-order comparisons of related bacteria is dependent on the model. Therefore, it is highly important to establish reliable assumptions before inferring its magnitude. RESULTS We investigate the patterns of dotplots between species of bacteria with the purpose of model selection in gene-order problems. We find several categories of data which can be explained by carefully weighing the contributions of reversals, transpositions, symmetrical reversals, single gene transpositions and single gene reversals. We also derive method of moments distance estimates for some previously uncomputed cases, such as symmetrical reversals, single gene reversals and their combinations, as well as the single gene transpositions edit distance.

Combinatorial properties of permutation tableaux

We give another construction of a permutation tableau from its corresponding permutation and construct a permutation-preserving bijection between 1-hinge and 0-hinge tableaux. We also consider certain alignment and crossing statistics on permutation tableaux that have previously been shown to be equidistributed by mapping them to patterns in related permutations. We give two direct maps on tableaux that prove the equidistribution of those statistics by exchanging some statistics and preserving the rest. Finally, we enumerate some sets of permutations that are restricted both by pattern avoidance and by certain parameters of their associated permutation tableaux.

Expected number of breakpoints after t random reversals in genomes with duplicate genes

In comparative genomics, one wishes to deduce the evolutionary distance between different species by studying their genomes. Using gene order information, we seek the number of times the gene order has changed between two species. One approach is to compute the method of moments estimate of this edit distance from a measure of dissimilarity called the breakpoint measure. In this paper, we extend the formulae and bounds of this estimate on gene permutations to genomes with duplicate genes.

The Freshman's Approach to Conway's Napkin Problem

In the March 2006 issue of the MONTHLY, Claesson and Petersen gave a thorough solution to Conways napkin problem. The problem is the following: Assume that n mathematicians arrive in random order at a conference dinner with a circular table, and that the napkins are placed exactly halfway between the plates so that the guests do not know whether they are supposed to use the right or the left napkin. Each guest prefers these napkins with probabilities p and 1-p, respectively, and tries her preferred alternative before trying the other, if the preferred napkin has been taken. Which proportion of guests is expected to sit down at a place where both adjacent napkins have been taken and thus be without a napkin? Claesson and Petersen use a system of generating functions to compute both the expectation and the variance of this proportion and to address similar questions, for instance regarding the number of guests who get a napkin though not the preferred one. However, these expectations can also be computed using purely elementary methods, such as the binomial theorem. We present the freshmans approach to the napkin problem and related problems, for instance the one with French diners mentioned, but not solved, by Claesson and Petersen.