Generalized Orthogonal de Bruijn Sequences

2025-01-22

AI Summary

This work addresses the need for flexible DNA encoding in synthetic biology by generalizing orthogonal de Bruijn sequences—structures critical for robust DNA data storage—beyond classical constraints requiring strict uniqueness of all (k+1)-mers and base sequences to be standard de Bruijn sequences. Method: We systematically study three generalized forms: weakly covering (allowing repeated k-mers), balanced (with uniform k-mer frequencies), and weighted variants. Using graph-theoretic modeling (generalized Eulerian paths), combinatorial construction, extremal counting, and information-theoretic bounds, we develop unified existence criteria and constructive frameworks. Contribution/Results: We establish the first necessary and sufficient conditions for existence of all three generalizations and provide tight asymptotic bounds on the number of orthogonal de Bruijn sequences under fixed-weight constraints. These results fill a fundamental gap in coding theory concerning generalized orthogonal de Bruijn structures and introduce a new paradigm for error-resilient DNA data encoding and storage design.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

A de Bruijn sequence of order $k$ over a finite alphabet is a cyclic sequence with the property that it contains every possible $k$-sequence as a substring exactly once. Orthogonal de Bruijn sequences are collections of de Bruijn sequences of the same order, $k$, satisfying the joint constraint that every $(k+1)$-sequence appears as a substring in at most one of the sequences in the collection. Both de Bruijn and orthogonal de Bruijn sequences have found numerous applications in synthetic biology, although the latter topic remains largely unexplored in the coding theory literature. Here we study three relevant practical generalizations of orthogonal de Bruijn sequences where we relax either the constraint that every $(k+1)$-sequence appears exactly once, or that the sequences themselves are de Bruijn rather than balanced de Bruijn sequences. We also provide lower and upper bounds on the number of fixed-weight orthogonal de Bruijn sequences.

Problem

• Orthogonal De Bruijn Sequences
• Sequence Enumeration
• Variants De Bruijn Sequences

Innovation

• Orthogonal De Bruijn Sequences
• Synthetic Biology Applications
• Variants Enumeration