Copy number variants (CNVs) are a class of structural variants that may involve complex genomic rearrangements (CGRs) and are hypothesized to have additional mutations around their breakpoints. Understanding the mechanisms underlying CNV formation is fundamental for understanding the repair and mutation mechanisms in cells, thereby shedding light on evolution, genomic disorders, cancer, and complex human traits. In this study, we used data from the 1000 Genomes Project to analyze hundreds of loci harboring heterozygous germline deletions in the subjects NA12878 and NA19240. By utilizing synthetic long-read data (longer than 2 kbp) in combination with high coverage short-read data and, in parallel, by comparing with parental genomes, we interrogated the phasing of these deletions with the flanking tens of thousands of heterozygous SNPs and indels. We found that the density of SNPs/indels flanking the breakpoints of deletions (in-phase variants) is approximately twice as high as the corresponding density for the variants on the haplotype without deletion (out-of-phase variants). This fold change was even larger for the subset of deletions with signatures of replication-based mechanism of formation. The allele frequency (AF) spectrum for deletions is enriched for rare events; and the AF spectrum for in-phase SNPs is shifted toward this deletion spectrum, thus offering evidence consistent with the concomitance of the in-phase SNPs/indels with the deletion events. These findings therefore lend support to the hypothesis that the mutational mechanisms underlying CNV formation are error prone. Our results could also be relevant for resolving mutation-rate discrepancies in human and to explain kataegis.

We consider normal state properties, the pairing instability temperature, and the structure of the pairing gap in electron-doped cuprates. We assume that the pairing is mediated by collective spin excitations, with antiferromagnetism emerging with the appearance of hot spots. We use a low-energy spin-fermion model and Eliashberg theory up to two-loop order. We justify ignoring vertex corrections by extending the model to N >> 1 fermionic flavors, with 1/N playing the role of a small Eliashberg parameter. We argue, however, that it is still necessary to solve coupled integral equations for the frequency dependent fermionic and bosonic self-energies, both in the normal and superconducting state. Using the solution of the coupled equations, we find an onset of d−wave pairing at Tc ∼ 30 K, roughly three times larger than the one obtained previously [P. Krotkov and A. Chubukov, Phys. Rev. B 74, 014509 (2006)], where it was assumed that the equations for fermionic and bosonic self-energies decouple in the normal state. To obtain the momentum and frequency dependent d-wave superconducting gap, ∆( kF , ωn), we derive and solve the non-linear gap equation together with the modified equation for the bosonic self energy which below Tc also depends on ∆( kF , ωn). We find that ∆( kF , ωn) is a non-monotonic function of momentum along the Fermi surface, with its node along the zone diagonal and its maximum some distance away from it. We obtain 2∆max(T → 0)/Tc ∼ 4. We argue that the value of Tc, the non-monotonicity of the gap, and 2∆max/Tc ratio are all in good agreement with the experimental data on electron-doped cuprates.