"Inverting" Caspar-Klug design rules for programmable assembly of size-controlled minimal-surface assemblies*

As first suggested by Caspar and Klug, many viruses assemble icosahedral shells (capsids) because the high symmetry of the icosahedron enables economical assembly – enclosing a large volume with relatively few distinct protein subunit types. We generalize this design principle to triply-periodic polyhedra, mesoporous structures approximating cubic minimal surfaces. We propose and investigate an extension of the Caspar-Klug symmetry principles for viral capsid assembly to the programmable assembly of size-controlled triply-periodic polyhedra, discrete variants of the Primitive, Diamond, and Gyroid cubic minimal surfaces. Inspired by a recent class of programmable DNA origami triangular colloids, we demonstrate that the economy of design in these crystalline assemblies -- in terms of the growth of the number of distinct particle species required with the increased size-scale (e.g. periodicity) -- is comparable to viral shells. We further test the role of geometric specificity in these assemblies via dynamical assembly simulations, which show that conditions for simultaneously efficient and high-fidelity assembly require an intermediate degree of flexibility of local angles and lengths in programmed assembly. Off-target misassembly occurs via incorporation of a variant of disclination defects, generalized to the case of hyperbolic crystals. The possibility of these topological defects is a direct consequence of the very same symmetry principles that underlie the economical design, exposing a basic tradeoff between design economy and fidelity of programmable, size controlled assembly.