DNA storage: how uneven sampling hides missing sequences

Selective file retrieval still involves sampling

Organick and colleagues' 2018 experiment stored 35 files, over 200 MB, in 13,448,372 distinct sequences. PCR primer targets selected a file; strand addresses ordered its fragments. Reads were clustered, reconstructed and decoded with Reed-Solomon correction. This was file selection, not direct electronic access to one byte. [1, pp. 242-245]

The often-cited 5x result is the median across files of coverage needed after subsampling usable reads; individual files needed 4x to 14x. Counting unusable reads raises that median to 6.2x. It does not mean every strand was read five times. These historical results are not today's device specification. [1, p. 245]

DNA read depth therefore needs a denominator: intended sequence identities, the selected file or entire pool, and raw or usable reads. Repeated observations of a popular identity cannot stand in for a missing one.

Count unseen identities, not just total reads

Heckel and colleagues analyze unordered DNA storage with independent, uniform sampling and error-free reads. Their model exposes the loss from sequences never drawn, while explicitly excluding realistic read errors and unequal amplification. [2, sections 1-3 and 5]

For a finite pool, let N be the fixed number of reads and p_i the probability that one read selects identity i. The probability of missing i on every draw is (1 - p_i)^N. Add the individual indicators to obtain:

E[U] = sum over i of (1 - p_i)^N

U counts identities observed zero times. Its expectation can be fractional, even though a single experiment loses an integer count. For M equally likely identities, p_i = 1/M and the expression becomes M(1 - 1/M)^N. With c = N/M and large M, this is approximately M exp(-c). The table uses the finite expression.

No independence between the missing-identity events is required for this expectation. In fact those events are dependent when N is fixed. Multiplying their individual presence probabilities would not give the exact probability that all identities were seen.

The same coverage, two very different pools

Both scenarios contain 10,000 intended identities. The uniform case assigns probability 0.0001 to each. The uneven case assigns 0.00001 to each of 1,000 rare identities and 0.00011 to each of 9,000 common identities. The masses sum to one: 0.01 + 0.99. Read count and pool size stay fixed; only the probability distribution changes.

At 10x overall coverage, a rare identity has only one expected read; a common identity has eleven. Averages hide that imbalance. At 20x, the uniform expectation is about 0.0000206 unseen identities, whereas the uneven case still leaves about 135.

An expectation below one is not a guarantee of completeness. The union bound gives P(U > 0) ≤ min(1, E[U]). For the uniform 10x case, 0.454 is an upper bound on that probability, not an equality. For the uneven cases here, the capped bound is one and uninformative. No laboratory confidence interval is attached to these fixed assumptions.

Unseen sequences need not mean an unrecoverable file

A decoder may reconstruct missing encoded fragments using redundancy. Conversely, observing an identity once does not establish that its sequence was read correctly. The toy model assumes perfect identity recognition and excludes substitutions, insertions, deletions and reconstruction errors. It cannot predict a decoder's success rate.

For engineering evaluation, record the raw-read budget, usable-read filter, coverage distribution, zero-read fraction, code parameters and byte-for-byte recovery result. A satisfactory average is insufficient evidence for any one of these checks.

Also separate logical density, expressed as payload bits per synthesized nucleotide with stated overheads, from physical density, expressed as payload bytes per stated mass or volume. A capacity claim does not establish the read budget, retrieval latency, retention lifetime or complete-system footprint. Compare designs using the same accounting boundary.

Continue with Reed-Solomon recovery guarantees, DNA data storage, oligonucleotide dropout and sequencing coverage.

Reproduce the calculation

The CC0 model uses Node.js built-ins only. Its JSON receipt contains all 12 rows, assumptions, runtime, generation time and source hash. There is no random seed: the results are expectations, not sampled trials.

node dna-coverage.mjs

Regression tests independently enumerate all 81 four-draw sequences for a three-identity weighted pool, then compare the weighted missing count with the formula. They also check zero reads, certain selection, zero-probability identities, invalid inputs and the published receipt. Source and tests are versioned in the project repository.

More realistic work would need measured sequence abundances, stage-specific losses, filtering and a specified decoder. This example isolates one reason a uniform assumption can be optimistic; it is not a reproduction of either paper or a newly discovered biological effect.

Primary sources

1. Organick et al. Random access in large-scale DNA data storage. Nature Biotechnology 36, 242-248 (2018). DOI: 10.1038/nbt.4079
2. Heckel, Shomorony, Ramchandran and Tse. Fundamental Limits of DNA Storage Systems. arXiv:1705.04732v1 (2017)

The Organick paper has a reference-order erratum, DOI 10.1038/nbt0718-660c. The inspected author-hosted PDF includes that correction.