On Achievable Rates for the Shotgun Sequencing Channel with Erasures

TL;DR

This work analyzes the Shotgun Sequencing Channel with Erasures (SSE($\delta$)), capturing base-call erasures in reads via a per-symbol erasure probability. It extends prior capacity results for the noiseless-read Shotgun Sequencing Channel by deriving an achievable-rate bound through a random code and a three-phase, typicality-based decoder that merges reads into islands. The main result provides a explicit lower bound on achievable rate: $R < (1- e^{-c(1-\delta)}) - (1-\delta)\left(e^{-c\left(1-\frac{1}{\bar{L}(1-\delta)}\right)} - e^{-c}\right)$, which reduces to the known capacity when $\delta=0$, and is supported by concentration lemmas for island formation and coverage. The findings illuminate how quality-score erasures degrade capacity and guide design considerations for DNA storage pipelines, while leaving open the development of tight converses and practical, efficient coding schemes for SSE.

Abstract

In shotgun sequencing, the input string (typically, a long DNA sequence composed of nucleotide bases) is sequenced as multiple overlapping fragments of much shorter lengths (called \textit{reads}). Modelling the shotgun sequencing pipeline as a communication channel for DNA data storage, the capacity of this channel was identified in a recent work, assuming that the reads themselves are noiseless substrings of the original sequence. Modern shotgun sequencers however also output quality scores for each base read, indicating the confidence in its identification. Bases with low quality scores can be considered to be erased. Motivated by this, we consider the \textit{shotgun sequencing channel with erasures}, where each symbol in any read can be independently erased with some probability $δ$. We identify achievable rates for this channel, using a random code construction and a decoder that uses typicality-like arguments to merge the reads.

Figures (3)

• Figure 1: The Shotgun Sequencing Channel with Erasures ($\mathsf{SSE}(\delta)$). The collection ${\cal \tilde{Y}} = \{\tilde{\underline{y}}_1, \tilde{\underline{y}}_2, \cdots, \tilde{\underline{y}}_{K} \}$ may be visualized as the output of the Shotgun Sequencing Channel ravi_coded_ssc, and ${{\cal Y}} = \{\underline{y}_{1}, \underline{y}_{2}, \cdots, \underline{y}_{K} \}$ is the output of $\mathsf{SSE}(\delta)$, after bits in each read are erased (indicated in bold/red) with probability $\delta$.
• Figure 2: The plot shows the rates from Theorem \ref{['thm:main']}, with $\bar{L}=1.75$, as the coverage depth $c$ varies, for $\delta=0, 0.05, 0.2,$ and $0.3$. We compare these with the capacity of the shotgun sequencing channel (denoted by SSC) from ravi_coded_ssc with read lengths $\bar{L}(1-\delta)\log n$.
• Figure 3: Illustration of a merge operation.

Theorems & Definitions (24)

• Theorem 1
• Remark 1
• Remark 2
• Definition 1: Length and Size of string
• Definition 2: Prefix and Suffix
• Definition 3: Compatibility, $l$-compatible strings and substring compatibility
• Definition 4: Merge of two strings
• Definition 5: True Successors, Ordering, and Overlaps
• Definition 6: Orderings, Islands, and True Islands
• Definition 7