Inference of chromosome-specific copy numbers using population haplotypes

<p>Abstract</p> <p>Background</p> <p>Using microarray and sequencing platforms, a large number of copy number variations (CNVs) have been identified in humans. In practice, because our human genome is a diploid, these platforms are limited to or more accurate for detecting total copy numbers rather than chromosome-specific copy numbers at each of the two homologous chromosomes. Nevertheless, the analysis of linkage disequilibrium (LD) between CNVs and SNPs indicates that distinct copy numbers often sit on their own background haplotypes.</p> <p>Results</p> <p>We propose new computational models for inferring chromosome-specific copy numbers by distinguishing background haplotypes of each copy number. The formulated problems are shown to be NP-hard and approximation/heuristic algorithms are developed. Simulation indicates that our method is accurate and outperforms the existing approach. By testing the program in 60 parent-offspring trios, the inferred chromosome-specific copy numbers are highly consistent with the law of Mendelian inheritance. The distributions of copy numbers at chromosomal level are provided for 270 individuals in three HapMap panels.</p> <p>Conclusions</p> <p>The estimation of chromosome-specific copy numbers using microarray or sequencing platforms was often confounded by a number of factors. This study showed that the integration of background haplotypes is able to improve the accuracies of copy number estimation at chromosome level, especially for the CNVs having strong LD with SNPs in proximity.</p

A Numerical Approach to Solving an Inverse Heat Conduction Problem Using the Levenberg-Marquardt Algorithm

This chapter is intended to provide a numerical algorithm involving the combined use of the Levenberg-Marquardt algorithm and the Galerkin finite element method for estimating the diffusion coefficient in an inverse heat conduction problem (IHCP). In the present study, the functional form of the diffusion coefficient is an unknown priori. The unknown diffusion coefficient is approximated by the polynomial form and the present numerical algorithm is employed to find the solution. Numerical experiments are presented to show the efficiency of the proposed method

Structure of pressure-gradient-driven current singularity in ideal magnetohydrodynamic equilibrium

Singular currents typically appear on rational surfaces in non-axisymmetric ideal magnetohydrodynamic equilibria with a continuum of nested flux surfaces and a continuous rotational transform. These currents have two components: a surface current (Dirac $\delta$-function in flux surface labeling) that prevents the formation of magnetic islands and an algebraically divergent Pfirsch--Schl\"uter current density when a pressure gradient is present across the rational surface. At flux surfaces adjacent to the rational surface, the traditional treatment gives the Pfirsch--Schl\"uter current density scaling as $J\sim1/\Delta\iota$, where $\Delta\iota$ is the difference of the rotational transform relative to the rational surface. If the distance $s$ between flux surfaces is proportional to $\Delta\iota$, the scaling relation $J\sim1/\Delta\iota\sim1/s$ will lead to a paradox that the Pfirsch--Schl\"uter current is not integrable. In this work, we investigate this issue by considering the pressure-gradient-driven singular current in the Hahm\textendash Kulsrud\textendash Taylor problem, which is a prototype for singular currents arising from resonant magnetic perturbations. We show that not only the Pfirsch--Schl\"uter current density but also the diamagnetic current density are divergent as $\sim1/\Delta\iota$. However, due to the formation of a Dirac $\delta$-function current sheet at the rational surface, the neighboring flux surfaces are strongly packed with $s\sim(\Delta\iota)^{2}$. Consequently, the singular current density $J\sim1/\sqrt{s}$, making the total current finite, thus resolving the paradox