# Generating information-dense promoter sequences with optimal string packing

## TL;DR

This paper introduces a computational method to design DNA sequences with densely packed binding sites, enabling efficient creation of synthetic promoters for studying gene regulation.

## Contribution

The novel contribution is a provably optimal algorithm for packing DNA binding sites into short sequences using integer linear programming.

## Key findings

- The nucleotide String Packing Problem is NP-hard and can be reduced to an Orienteering Problem for efficient solving.
- The method packs 20–100 binding sites into 50–300 base pair sequences in seconds with provable optimality.
- The approach allows designing bacterial promoters with fixed sequence elements and controls binding site usage frequency.

## Abstract

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 sets 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. As an example, we then show how adding additional constraints, like the inclusion of sequence elements with fixed positions, allows for the design of bacterial promoters. 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 could help interrogate how complex binding site arrangements impact either gene expression or biomolecular mechanisms in varied cellular contexts.

The way protein binding sites are arranged on DNA can influence the regulation and transcription of downstream genes. Areas with a high concentration of binding sites can enable complex interplay between transcription factors, a feature that is exploited by natural promoters. However, designing synthetic promoters that contain dense arrangements of binding sites is a challenge. The task involves overlapping many binding sites, each typically about 10 nucleotides long, within a constrained sequence area, which becomes increasingly difficult as sequence length decreases and binding site variety increases. We introduce an approach to design nucleotide sequences with optimally packed protein binding sites, which we call the nucleotide String Packing Problem (SPP). We show that the SPP can be solved efficiently using integer linear programming to identify the densest arrangements of binding sites for a specified sequence length. We show how adding additional constraints, like the inclusion of sequence elements with fixed positions, allows for the design of bacterial promoters. The presented approach enables the rapid design and study of nucleotide sequences with complex, dense binding site architectures.