Coverage processes in physical mapping by anchoring random clones

Coverage theories for metagenomic DNA sequencing based on a generalization of Stevens' theorem

Metagenomic project design has relied variously upon speculation, semi-empirical and ad hoc heuristic models, and elementary extensions of single-sample Lander–Waterman expectation theory, all of which are demonstrably inadequate. Here, we propose an approach based upon a generalization of Stevens’ Theorem for randomly covering a domain. We extend this result to account for the presence of multiple species, from which are derived useful probabilities for fully recovering a particular target microbe of interest and for average contig length. These show improved specificities compared to older measures and recommend deeper data generation than the levels chosen by some early studies, supporting the view that poor assemblies were due at least somewhat to insufficient data. We assess predictions empirically by generating roughly 4.5 Gb of sequence from a twelve member bacterial community, comparing coverage for two particular members, Selenomonas artemidis and Enterococcus faecium, which are the least (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim $$\end{document}3 %) and most (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim $$\end{document}12 %) abundant species, respectively. Agreement is reasonable, with differences likely attributable to coverage biases. We show that, in some cases, bias is simple in the sense that a small reduction in read length to simulate less efficient covering brings data and theory into essentially complete accord. Finally, we describe two applications of the theory. One plots coverage probability over the relevant parameter space, constructing essentially a “metagenomic design map” to enable straightforward analysis and design of future projects. The other gives an overview of the data requirements for various types of sequencing milestones, including a desired number of contact reads and contig length, for detection of a rare viral species.

Coverage statistics for sequence census methods

Background

We study the statistical properties of fragment coverage in genome sequencing experiments. In an extension of the classic Lander-Waterman model, we consider the effect of the length distribution of fragments. We also introduce a coding of the shape of the coverage depth function as a tree and explain how this can be used to detect regions with anomalous coverage. This modeling perspective is especially germane to current high-throughput sequencing experiments, where both sample preparation protocols and sequencing technology particulars can affect fragment length distributions.

Results

Under the mild assumptions that fragment start sites are Poisson distributed and successive fragment lengths are independent and identically distributed, we observe that, regardless of fragment length distribution, the fragments produced in a sequencing experiment can be viewed as resulting from a two-dimensional spatial Poisson process. We then study the successive jumps of the coverage function, and show that they can be encoded as a random tree that is approximately a Galton-Watson tree with generation-dependent geometric offspring distributions whose parameters can be computed.

Conclusions

We extend standard analyses of shotgun sequencing that focus on coverage statistics at individual sites, and provide a null model for detecting deviations from random coverage in high-throughput sequence census based experiments. Our approach leads to explicit determinations of the null distributions of certain test statistics, while for others it greatly simplifies the approximation of their null distributions by simulation. Our focus on fragments also leads to a new approach to visualizing sequencing data that is of independent interest.

Theories and Applications for Sequencing Randomly Selected Clones

Theory is developed for the process of sequencing randomly selected large-insert clones. Genome size, library depth, clone size, and clone distribution are considered relevant properties and perfect overlap detection for contig assembly is assumed. Genome-specific and nonrandom effects are neglected. Order of magnitude analysis indicates library depth is of secondary importance compared to the other variables, especially as clone size diminishes. In such cases, the well-known Poisson coverage law is a good approximation. Parameters derived from these models are used to examine performance for the specific case of sequencing random human BAC clones. We compare coverage and redundancy rates for libraries possessing uniform and nonuniform clone distributions. Results are measured against data from map-based human-chromosome-2 sequencing. We conclude that the map-based approach outperforms random clone sequencing, except early in a project. However, simultaneous use of both strategies can be beneficial if a performance-based estimate for halting random clone sequencing is made. Results further show that the random approach yields maximum effectiveness using nonbiased rather than biased libraries.

The effect of nonhomogeneous clone length distribution on the progress of an STS mapping project

We provide both theoretical and simulation results on the progress of an STS mapping project in the presence of clone length inhomogeneity. For an example in which the genome comprises alternating regions of clones with short and long average length, the main conclusion is that the efficiency of the project is clearly decreased in the presence of such inhomogeneity. The case of deterministic clone length gives the worst progress. The general simulation algorithm we propose shows that strategies that space the anchors as regularly as possible do best: fewer contigs of larger average length are expected. The simulation algorithm can be used to study many statistical properties of the progress of any anchoring project.

An overview on the distribution of word counts in Markov chains

In this paper, we give an overview about the different results existing on the statistical distribution of word counts in a Markovian sequence of letters. Results concerning the number of overlapping occurrences, the number of renewals and the number of clumps will be presented. Counts of single words and also multiple words are considered. Most of the results are approximations as the length of the sequence tends to infinity. We will see that Gaussian approximations switch to (compound) Poisson approximations for rare words. Modeling DNA sequences or proteins by stationary Markov chains, these results can be used to study the statistical frequency of motifs in a given sequence.