Complex DNA Synthesis Sequences
Boaz Moav, Ryan Gabrys, Eitan Yaakobi
TL;DR
This work develops a comprehensive theoretical framework for constrained DNA synthesis, introducing a complex synthesis model where each cycle selects from a restricted nucleotide subset of size $w$ from an alphabet of size $q$. It derives exact information-rate expressions using complex deletion-ball analysis, provides closed-form rate bounds, and demonstrates asymptotic behavior, including the notable limit $\lim_{q\to\infty} f(q,w) = \log_2(w+1)$. The authors also design dynamic-programming algorithms to compute optimal complex synthesis sequences (SCCS) for sets of strands and propose an approximation method to handle large collections, plus a parallel two-dimensional array model with a DP solution and connections to the shortest common supersequence problem. Collectively, these results unify prior synthesis models, quantify capacity under constraints, and lay groundwork for practical constrained-synthesis coding and large-scale DNA-array implementations.
Abstract
DNA-based storage offers unprecedented density and durability, but its scalability is fundamentally limited by the efficiency of parallel strand synthesis. Existing methods either allow unconstrained nucleotide additions to individual strands, such as enzymatic synthesis, or enforce identical additions across many strands, such as photolithographic synthesis. We introduce and analyze a hybrid synthesis framework that generalizes both approaches: in each cycle, a nucleotide is selected from a restricted subset and incorporated in parallel. This model gives rise to a new notion of a complex synthesis sequence. Building on this framework, we extend the information rate definition of Lenz et al. and analyze an analog of the deletion ball, defined and studied in this setting, deriving tight expressions for the maximal information rate and its asymptotic behavior. These results bridge the theoretical gap between constrained models and the idealized setting in which every nucleotide is always available. For the case of known strands, we design a dynamic programming algorithm that computes an optimal complex synthesis sequence, highlighting structural similarities to the shortest common supersequence problem. We also define a distinct two-dimensional array model with synthesis constraints over the rows, which extends previous synthesis models in the literature and captures new structural limitations in large-scale strand arrays. Additionally, we develop a dynamic programming algorithm for this problem as well. Our results establish a new and comprehensive theoretical framework for constrained DNA, subsuming prior models and setting the stage for future advances in the field.