The DNA Storage Channel: Capacity and Error Probability Bounds

Abstract

The DNA storage channel is considered, in which the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> Deoxyribonucleic acid (DNA) molecules comprising each codeword are stored without order, sampled <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> times with replacement, and then sequenced over a discrete memoryless channel. For a constant coverage depth <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M/N$ </tex-math></inline-formula> and molecule length scaling <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Theta (\log M)$ </tex-math></inline-formula> , lower (achievability) and upper (converse) bounds on the capacity of the channel, as well as a lower (achievability) bound on the reliability function of the channel are provided. Both the lower and upper bounds on the capacity generalize a bound which was previously known to hold only for the binary symmetric sequencing channel, and only under certain restrictions on the molecule length scaling and the crossover probability parameters. When specified to binary symmetric sequencing channel, these restrictions are completely removed for the lower bound and are significantly relaxed for the upper bound in the high-noise regime. The lower bound on the reliability function is achieved under a universal decoder, and reveals that the dominant error event is that of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">outage</i> – the event in which the capacity of the channel induced by the DNA molecule sampling operation does not support the target rate.