Structural and Compositional Complexities of Hierarchical Self-Assembly: a Hypergraph Approach
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909947069988864 |
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| author | Tkachenko, Alexei V. |
| author_facet | Tkachenko, Alexei V. |
| contents | Programmable self-assembly enables the construction of complex molecular, supramolecular, and crystalline architectures from well-designed building blocks. We introduce a hypergraph-based formalism, Blocks & Bonds (B&B), that generalizes classical chemical graph theory by incorporating directed and multicolored interactions, internal symmetries, and hierarchical organization. Within this framework, we develop the Structure Code (SC), a compact and versatile language for describing self-assembled architectures. We define a Kolmogorov-style Structural Complexity as the total information content of SC, obtained through its tokenization and Shannon information assignment. Complementing this encoding-based measure, we introduce a much simpler quantity, the Compositional Complexity, which depends only on the number and cumulative usage of block and bond types in the construction set. A central result of this work is a strong empirical correlation between the token-based Structural Complexity and the Compositional Complexity across all examined systems. Owing to this agreement, the Compositional Complexity emerges as the most practical and broadly applicable measure: it is easy to compute, requires no explicit encoding, and yet closely tracks the actual information content of structurally diverse architectures. Applications to molecular systems (ethylene glycol, glucose), DNA-origami lattices, and crystalline assemblies show that B\&B hypergraphs provide a unified, scalable, and information-efficient representation of structural organization, naturally capturing symmetry, modularity, and stereochemistry. This framework establishes a quantitative foundation for complexity-aware classification and inverse design of programmable matter. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_26449 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structural and Compositional Complexities of Hierarchical Self-Assembly: a Hypergraph Approach Tkachenko, Alexei V. Soft Condensed Matter Mesoscale and Nanoscale Physics Mathematical Physics Programmable self-assembly enables the construction of complex molecular, supramolecular, and crystalline architectures from well-designed building blocks. We introduce a hypergraph-based formalism, Blocks & Bonds (B&B), that generalizes classical chemical graph theory by incorporating directed and multicolored interactions, internal symmetries, and hierarchical organization. Within this framework, we develop the Structure Code (SC), a compact and versatile language for describing self-assembled architectures. We define a Kolmogorov-style Structural Complexity as the total information content of SC, obtained through its tokenization and Shannon information assignment. Complementing this encoding-based measure, we introduce a much simpler quantity, the Compositional Complexity, which depends only on the number and cumulative usage of block and bond types in the construction set. A central result of this work is a strong empirical correlation between the token-based Structural Complexity and the Compositional Complexity across all examined systems. Owing to this agreement, the Compositional Complexity emerges as the most practical and broadly applicable measure: it is easy to compute, requires no explicit encoding, and yet closely tracks the actual information content of structurally diverse architectures. Applications to molecular systems (ethylene glycol, glucose), DNA-origami lattices, and crystalline assemblies show that B\&B hypergraphs provide a unified, scalable, and information-efficient representation of structural organization, naturally capturing symmetry, modularity, and stereochemistry. This framework establishes a quantitative foundation for complexity-aware classification and inverse design of programmable matter. |
| title | Structural and Compositional Complexities of Hierarchical Self-Assembly: a Hypergraph Approach |
| topic | Soft Condensed Matter Mesoscale and Nanoscale Physics Mathematical Physics |
| url | https://arxiv.org/abs/2509.26449 |