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

TL;DR

The paper studies the minimum number of length-$m$ DNA labels needed to achieve the maximal labeling capacity $\log_2 q$ by translating the problem into maximizing the edge count $\gamma(q,d)$ of path-unique subgraphs of the $d$-dimensional de Bruijn graph $\mathcal{B}_{q,d}$. It provides two constructions to lower-bound $\gamma(q,d)$ and derives a general upper bound via the Walk-overlap function $\eta(q,d,k)$, establishing explicit bounds and asymptotics. A key result is that $\lim_{d\to\infty} \gamma(q,d)/q^{d+1}=1$, demonstrated through connections to universal hitting sets and minimum decycling sets, alongside analytical bounds and numerical comparisons. These findings advance understanding of DNA labeling capacity limits and offer guidance for designing high-capacity labeling schemes in molecular biology and related domains, by clarifying how many length-$m$ labels are needed to reach the theoretical capacity $\log_2 q$.

Abstract

DNA labeling is a tool in molecular biology and biotechnology to visualize, detect, and study DNA at the molecular 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 non-zero 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 $\log_2q$ 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. We draw new connections to existing literature that let us prove an asymptotic result as the label length tends to infinity.

Figures (7)

• Figure 1: An optimal path unique graph over $\Sigma_4=\{A,C,G,T\}$.
• Figure 2: The edges of part \ref{['const1-cond1']} of Construction \ref{['constr:1']} for $q=3$.
• Figure 3: The graph $\mathcal{G}^2_{5}$ from Construction \ref{['constr:2']}. For compactness, each vertex $(x_1, x_2) \in \Sigma_q^2$ is labeled by $x_1x_2$, e.g., $(0, 1)$ by $01$. Edges satisfying Condition \ref{['const2-cond1']} are blue, edges satisfying Condition \ref{['const2-cond2']} are red, edges satisfying Condition \ref{['const2-cond3']} are black, edges satisfying Condition \ref{['const2-cond4']} are green, and edges satisfying Condition \ref{['const2-cond5']} are purple.
• Figure 4: The de Bruijn graph $\mathcal{B}_{2, 3}$ with the cycles corresponding to necklaces highlighted by color; 000 in orange, 001 in green, 011 in blue, and 111 in pink. The four nodes with dashed circles form a minimum decycling set of the graph.
• Figure 5: Visualization of the Mykkeltveit V-set and its extension for the binary alphabet. The V-sets for $2\leq d \leq 16$ are extended until the length of the longest avoiding walk is $d+1$ (left figure). This increases $d$ but preserves the relative cardinality of the set (right figure). Since the relative cardinality of the V-set shrinks to zero as $d\to \infty$, so does the relative cardinality of the extended set.

Theorems & Definitions (45)

• Definition 1: The labeling model hanania2023capacity
• Example 1
• Definition 2: de Bruijn graphs
• Definition 3: Path unique graph
• Theorem 1
• Corollary 1
• Theorem 2: see zhan2012Extremal
• Lemma 1
• Lemma 2
• proof