Coverage (or depth) in DNA sequencing is the number of reads that include a given nucleotide in the reconstructed sequence. Deep sequencing refers to the general concept of aiming for high number of replicate reads of each region of a sequence.
Even though the sequencing accuracy for each individual nucleotide is very high, the very large number of nucleotides in the genome means that if an individual genome is only sequenced once, there will be a significant number of sequencing errors. Furthermore, many positions in a genome contain rare single-nucleotide polymorphisms (SNPs). Hence to distinguish between sequencing errors and true SNPs, it is necessary to increase the sequencing accuracy even further by sequencing individual genomes a large number of times.
The term "ultra-deep" can sometimes also refer to higher coverage (>100-fold), which allows for detection of sequence variants in mixed populations. In the extreme, error-corrected sequencing approaches such as Maximum-Depth Sequencing can make it so that coverage of a given region approaches the throughput of a sequencing machine, allowing coverages of >10^8 
Deep sequencing of transcriptomes, also known as RNA-Seq, provides both the sequence and frequency of RNA molecules that are present at any particular time in a specific cell type, tissue or organ. Counting the number of mRNAs that are encoded by individual genes provides an indicator of protein-coding potential, a major contributor to phenotype. Improving methods for RNA sequencing is an active area of research both in terms of experimental and computational methods.
The average coverage for a whole genome can be calculated from the length of the original genome (G), the number of reads (N), and the average read length (L) as . For example, a hypothetical genome with 2,000 base pairs reconstructed from 8 reads with an average length of 500 nucleotides will have 2× redundancy. This parameter also enables one to estimate other quantities, such as the percentage of the genome covered by reads (sometimes also called coverage). A high coverage in shotgun sequencing is desired because it can overcome errors in base calling and assembly. The subject of DNA sequencing theory addresses the relationships of such quantities.
Sometimes a distinction is made between sequence coverage and physical coverage. Where sequence coverage is the average number of times a base is read, physical coverage is the average number of times a base is read or spanned by mate paired reads.
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