Building Polyhedra by Self-Assembly: Theory and Experiment

Abstract

We investigate the utility of a mathematical framework based on discrete geometry to model biological and synthetic self-assembly. Our primary biological example is the self-assembly of icosahedral viruses; our synthetic example is surface-tension-driven self-folding polyhedra. In both instances, the process of self-assembly is modeled by decomposing the polyhedron into a set of partially formed intermediate states. The set of all intermediates is called the configuration space, pathways of assembly are modeled as paths in the configuration space, and the kinetics and yield of assembly are modeled by rate equations, Markov chains, or cost functions on the configuration space. We review an interesting interplay between biological function and mathematical structure in viruses in light of this framework. We discuss in particular: (i) tiling theory as a coarse-grained description of all-atom models; (ii) the building game-a growth model for the formation of polyhedra; and (iii) the application of these models to the self-assembly of the bacteriophage MS2. We then use a similar framework to model self-folding polyhedra. We use a discrete folding algorithm to compute a configuration space that idealizes surface-tension-driven self-folding and analyze pathways of assembly and dominant intermediates. These computations are then compared with experimental observations of a self-folding dodecahedron with side 300 μm. In both models, despite a combinatorial explosion in the size of the configuration space, a few pathways and intermediates dominate self-assembly. For self-folding polyhedra, the dominant intermediates have fewer degrees of freedom than comparable intermediates, and are thus more rigid. The concentration of assembly pathways on a few intermediates with distinguished geometric properties is biologically and physically important, and suggests deeper mathematical structure.

AMS Fall Sectional Sampler

“…However, the dodecahedron has 43,380 nets, and, to my surprise and delight, simple heuristics along with our computations revealed the best nets in the lab [4]. Since then our work has evolved into a study of the pathways of self-assembly [3]. This has required some surprisingly sophisticated mathematics.…”

Self‐Folding Using Capillary Forces

“…In terms of design criteria for capillary self‐folding of polyhedra, Gracias and Menon and coworkers discovered optimal self‐folding nets using a combination of experiments and mathematical analyses . They found that high yielding self‐folding of polyhedra occurs with compact nets with low radius of gyration ( R g ) and with nets featuring high secondary neighbors or vertex connections (secondary panels are those not directly connected by a folding hinge but that can interact with an aligning hinge) .…”

“…Importantly, intermediates that were more rigid with fewer degrees of freedom were preferred. Collectively, the unraveling of such rules greatly aids the problem of inverse design of complex structures by self‐folding . The optimal use of nearest neighbors and compact nets can enable defect tolerance and error correction .…”

“…[61] Subsequent studies in self-aligning liquid hinges deposited at the periphery of polyhedra enabled design of error tolerant nets and control over folding pathways. [93,[114][115][116] It is noteworthy that multiple arrangements and connections of polygons on nets are possible while folding a polyhedron. The number of nets increases exponentially with the number of faces of the polyhedron.…”

“…In terms of design criteria for capillary self-folding of polyhedra, Gracias and Menon and coworkers discovered optimal self-folding nets using a combination of experiments and mathematical analyses. [93,[114][115][116] They found that high yielding selffolding of polyhedra occurs with compact nets with low radius Adv. Mater.…”