Generating information-dense nucleotide sequences with optimal string packing.

Dense arrangements of binding sites within nucleotide sequences can collectively influence downstream transcription rates or initiate biomolecular interactions. For example, natural promoter regions can harbor many overlapping transcription factor binding sites that influence the rate of transcription initiation. Despite the prevalence of overlapping binding sites in nature, rapid design of nucleotide sequences with many overlapping sites remains a challenge. Here, we show that this is an NP-hard problem, coined here as the nucleotide String Packing Problem (SPP). We then introduce a computational technique that efficiently assembles sets of DNA-protein binding sites into dense, contiguous stretches of double-stranded DNA. For the efficient design of nucleotide sequences spanning hundreds of base pairs, we reduce the SPP to an Orienteering Problem with integer distances, and then leverage modern integer linear programming solvers. Our method optimally packs libraries of 20-100 binding sites into dense nucleotide arrays of 50-300 base pairs in 0.05-10 seconds. Unlike approximation algorithms or meta-heuristics, our approach finds provably optimal solutions. We demonstrate how our method can generate large sets of diverse sequences suitable for library generation, where the frequency of binding site usage across the returned sequences can be controlled by modulating the objective function. The nucleotide string packing approach we present can accelerate the design of sequences with complex DNA-protein interactions. When used in combination with synthesis and high-throughput screening, this design strategy can help interrogate how complex binding site arrangements impact either gene expression or biomolecular mechanisms in varied cellular contexts.