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Building polyhedra by self-assembly: theory and experiment.

Ryan Kaplan1, Joseph Klobušický, Shivendra Pandey

  • 1Brown University.

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Summary

A mathematical framework using discrete geometry models biological and synthetic self-assembly, from viruses to folding polyhedra. This approach reveals dominant pathways and intermediates, highlighting geometric properties crucial for self-assembly processes.

Keywords:
Virusmicrofabricationnanotechnologyorigamiself-foldingviral tiling theory

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Area of Science:

  • Biophysics
  • Mathematical Biology
  • Materials Science

Background:

  • Self-assembly is fundamental to biological systems and synthetic materials.
  • Modeling self-assembly requires understanding complex intermediate states and pathways.
  • Discrete geometry offers a novel lens for analyzing these processes.

Purpose of the Study:

  • To investigate a discrete geometry framework for modeling biological and synthetic self-assembly.
  • To analyze the interplay between mathematical structure and biological function in viral self-assembly.
  • To compare computational models with experimental data for self-folding polyhedra.

Main Methods:

  • Decomposition of polyhedra into intermediate states to define configuration spaces.
  • Modeling assembly pathways as paths within these configuration spaces.
  • Utilizing rate equations, Markov chains, cost functions, and discrete folding algorithms.

Main Results:

  • A few dominant pathways and intermediates emerge despite the combinatorial complexity of configuration spaces.
  • For self-folding polyhedra, dominant intermediates exhibit increased rigidity due to fewer degrees of freedom.
  • The framework successfully models both viral (icosahedral viruses, bacteriophage MS2) and synthetic (self-folding dodecahedron) systems.

Conclusions:

  • Discrete geometry provides a powerful tool for understanding the fundamental principles governing self-assembly.
  • Dominant intermediates with distinct geometric properties are key to efficient self-assembly.
  • This mathematical approach reveals underlying structures with significant biological and physical implications.