Capacity of Frequency-based Channels: Encoding Information in Molecular Concentrations

We consider a molecular channel, in which messages are encoded into the frequency of molecules in a pool, and whose output during reading time is a noisy version of the input frequencies, obtained by sampling with replacement from the pool. We tightly characterize the capacity of this channel using upper and lower bounds, when the number of molecules in the pool is constrained. We apply this result to the DNA storage channel in a accurately defined short-molecule regime, and show that even though the capacity of this channel is technically zero, it can still achieve very large information density.