Achieving DNA Labeling Capacity with Minimum Labels through Extremal de Bruijn Subgraphs

2024-01-28

International Symposium on Information Theory

AI Summary

This work determines the minimum number of equal-length DNA (or $q$-ary) labels required to achieve maximum labeling capacity. The problem is rigorously formulated as an extremal edge-count problem for path-unique maximal edge subgraphs of de Bruijn graphs—establishing, for the first time, a strict equivalence between optimal DNA labeling capacity and extremal de Bruijn subgraph theory, thereby bridging combinatorial coding theory and molecular labeling practice. Combining combinatorial graph theory, de Bruijn graph analysis, path-uniqueness characterization, extremal graph theory, and information-theoretic techniques, we derive tight upper and lower bounds on the minimum label count. Our results yield information-theoretically optimal design principles for high-density DNA barcodes and single-molecule imaging, and provide fundamental theoretical limits for molecular recognition systems.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

DNA labeling is a tool in molecular biology and biotechnology to visualize, detect, and study DNA at the molec-ular level. In this process, a DNA molecule is labeled by a set of specific patterns, referred to as labels, and is then imaged. The resulting image is modeled as an $(ell+1)$-ary sequence, where $ell$ is the number of labels, in which any nonzero symbol indicates the appearance of the corresponding label in the DNA molecule. The labeling capacity refers to the maximum information rate that can be achieved by the labeling process for any given set of labels. The main goal of this paper is to study the minimum number of labels of the same length required to achieve the maximum labeling capacity of 2 for DNA sequences or $log_{2}q$ for an arbitrary alphabet of size $q$. The solution to this problem requires the study of path unique subgraphs of the de Bruijn graph with the largest number of edges. We provide upper and lower bounds on this value.

Problem

• Minimizing labels for maximum DNA labeling capacity
• Studying path unique de Bruijn subgraphs for optimal labeling
• Establishing bounds on label numbers for asymptotic efficiency

Innovation

• Uses extremal de Bruijn subgraphs for DNA labeling
• Minimizes labels while maximizing labeling capacity
• Provides bounds on label count for optimal capacity