Pangene Graphs: Gene Content in Pangenomes

• Pangene graphs are formal graph-based models that represent gene content variations across genomes by integrating gene order, orientation, and copy-number changes.
• They combine gene family annotation, bidirected graph theory, and scalable alignment algorithms to construct compact, interpretable genomic representations.
• Key applications include annotation transfer, visualization of structural variants, and robust identification of core and accessory genes in both bacterial and eukaryotic cohorts.

A pangene graph is a formal graph-based model designed for the joint representation and comparative analysis of gene content variations across multiple genomes, particularly within large-scale eukaryotic pangenome cohorts. It combines approaches from gene family annotation, bidirected graph theory, and scalable alignment algorithms to encode gene presence, order, orientation, copy-number changes, and structural rearrangements as observed across all constituent assemblies. Pangene graphs generalize gene content graphs for use in both bacterial and eukaryotic settings and facilitate robust identification and visualization of core, accessory, and structurally variant genes (Li et al., 2024).

1. Formal Definition and Structure

Let $V$ denote a set of genes (orthology groups) extracted from a curated or clustered protein set. Each gene $v \in V$ can appear on either DNA strand, so the set of oriented genes is $X = V \times \{>,<\}$, where $>v$ denotes the forward orientation and $<v$ the reverse. Each assembled genome (or "contig" $t$) provides a walk in $X$, recording the in-order listing of oriented genes along that contig.

The canonical directed pangene gene content graph is $G_D = (X, E)$, where $E \subseteq X \times X$ and $x \to y$ whenever gene $v \in V$0 follows $v \in V$1 in at least one genome. This construction inherently satisfies a skew-symmetry property: if $v \in V$2, then the reversed complement $v \in V$3 is also present. The bidirected representation $v \in V$4 encodes each edge with explicit orientation at the endpoints. A further biedged form $v \in V$5 introduces separate “start” and “end” marker nodes for each gene and distinguishes internal from locus-adjacency edges, supporting bidirected transformations and efficient subgraph detection. Each genome (haplotype) corresponds to one or more contig walks in $v \in V$6, thus representing all traversed gene orderings in a compact structure (Li et al., 2024).

2. Construction Algorithm

The construction of a pangene graph follows a standardized multi-stage pipeline:

1. Protein Set Preparation: Start from a large, possibly redundant, set of protein sequences $v \in V$7 (canonical plus paralogs for human; e.g., GENCODE set). Reduce redundancy by clustering (e.g., CD-HIT at 95–98% identity).
2. Protein–Genome Alignment: For every genome assembly $v \in V$8 (haplotype- or isolate-resolved), align each protein $v \in V$9 against $X = V \times \{>,<\}$0 using a splice-aware aligner such as miniprot, capturing loci, orientation, and alignment scores robust to indels and frameshifts.
3. Gene Set Selection: For each gene, the “best” hit locus is determined per genome; overlapping or ambiguous regions are resolved so that, typically, orthologous genes across genomes collapse, while true paralogous copies are retained.
4. Edge Filtering and Graph Construction: After orthology/paralogy resolution, walks (ordered lists of oriented genes) are built for each contig. For every adjacent pair $X = V \times \{>,<\}$1, the directed edge $X = V \times \{>,<\}$2 is added to $X = V \times \{>,<\}$3 (with skew-symmetric complement). Heuristic edge filtering applied at this stage suppresses spurious connections due to misalignments or local deletions by comparison of alignment scores.
5. Core/Accessory Annotation: Genes are annotated as core if present in $X = V \times \{>,<\}$4 of genomes, and accessory otherwise.

This process yields a bidirected GFA-formatted pangene graph that efficiently encodes all observed local gene order, orientation, and copy-number configurations (Li et al., 2024).

3. Bibubbles: Subgraph Structures for Gene-Content Variation

A central analytic feature of pangene graphs is the detection of “bibubbles,” minimal subgraphs that capture gene content polymorphisms between genomes. Formally, for two oriented genes $X = V \times \{>,<\}$5, the bibubble structure $X = V \times \{>,<\}$6 satisfies:

• $X = V \times \{>,<\}$7, ensuring symmetric traversal on both DNA strands.
• Every $X = V \times \{>,<\}$8 is part of at least one $X = V \times \{>,<\}$9 walk.
• No smaller inlet or outlet exists that itself defines a nonempty bibubble (minimality).

Algorithmically, bibubble boundaries are located using cycle-equivalence classes in a contracted net graph representation $>v$0, followed by bounded breadth-first search to identify candidate boundaries, and reachability-based validation.

Bibubbles generalize classical superbubbles to bidirected graphs, thus capturing not only alternative order and copy-number but also inversions and strand switches inaccessible to bubble-detection algorithms designed for directed graphs (Li et al., 2024).

4. Computational Complexity and Performance

• Protein-to-genome alignment is the main computational bottleneck, scaling as $>v$1 with the number of proteins, genomes, and genome length.
• Graph construction and edge filtering require $>v$2 memory, with typical $>v$3.
• Bibubble detection runs in $>v$4 where $>v$5 is the number of branching nodes and $>v$6 a bound (typically $>v$7) on gene count per search, yielding near-linear behavior in practical settings.

Empirical timings include construction of the human pangenome (100 haplotypes, 19,421 proteins) in under a minute for graph build and less than a second for bibubble detection; comparable results hold for large bacterial panels (Li et al., 2024).

5. Biological and Methodological Applications

Pangene graphs support comprehensive gene content analysis at the full-genome scale in both microbial and eukaryotic contexts:

• Annotation transfer: Each node and path corresponds to explicit gene loci and order, facilitating cross-genome annotation projection and coordinate system definition.
• Visualization: The graph compactly displays alternative local gene orders, duplications, deletions, and orientation polymorphisms.
• Variant detection: Bibubbles pinpoint regions of structural and copy-number variability, as exemplified by HLA-DRB1 order switches, RHD copy-number variation, and CYP2D6/7 tandem duplication configurations in human assemblies.
• Quantitative characterization: The method offers gene-by-gene core/accessory status and exhibits close concordance with other established pangenome tools for gene family quantification (Panaroo, PPanGGOLiN).

The approach generalizes across species, providing a robust substrate for pangenome-wide association studies and scalable survey of eukaryotic gene content innovation (Li et al., 2024).

6. Theoretical Connections to Other Pangenome Models

Gene-level pangene graphs form one axis among a spectrum of pangenome graph models. In particular, the rigorous framework of Cicherski & Dojer relates string-graph-based representations, such as variation graphs and de Bruijn graphs, through a common axiomatization:

• Both model sets encode strings (genomes) as walks in node-labeled graphs.
• Variation graphs (singular, node labels of length 1) and de Bruijn graphs (node labels of length $>v$8) can be related via vertex splitting, edge rewiring, and merging, yielding equivalent representations under $>v$9-completeness and $<v$0-faithfulness constraints: both graphs admit canonical coordinate correspondences and so support consistent annotation and variant transfer (Cicherski et al., 14 Mar 2025).
• The Split $<v$1 Merge $<v$2 Collapse transformation efficiently constructs a minimal singular variation graph from a de Bruijn graph, just as pangene graph construction collapses redundant alignment and gene calls to yield a compact bidirected model.

A plausible implication is that pangene graphs, as bidirected gene content graphs, function as a special case of such string or variation graphs where labels correspond to full gene bodies rather than nucleotide substrings—maximizing interpretability at the expense of local nucleotide-level variation granularity (Li et al., 2024, Cicherski et al., 14 Mar 2025).

7. Computational Analysis and Open Problems

Evaluating sequence-to-pangenome graph similarity—such as the longest common subsequence (LCS) between a query and a pangene graph—can be reduced to longest path problems in directed acyclic graphs (DAGs). Four variants (LCS-SG, FGLCS-SG, MEMC, MSP) admit polynomial-time solutions via careful DAG constructions, though practical scalability remains limited by preprocessing costs (especially, $<v$3 all-pairs reachability computations for large graphs). Current and future work focuses on circumventing these bottlenecks via sparsity exploitation, indexing, and parallel computation (Li et al., 5 Feb 2026).

The analysis and construction methods underlying pangene graphs thus benefit from, and inform, broader algorithmic developments in graph-based pangenomics, extending from microbial core genome analysis to human pangenome variation at the gene and structural variant level.