Self-organized complex structures in nature, e.g., viral capsids, hierarchical biopolymers, and bacterial flagella, offer efficiency, adaptability, robustness, and multi-functionality. Can we program the self-assembly of three-dimensional (3D) complex structures using simple building blocks, and reach similar or higher level of sophistication in engineered materials? Here we present an analytic theory for the self-assembly of polyhedral nanoparticles (NPs) based on their crystal structures in non-Euclidean space. We show that the unavoidable geometrical frustration of these particle shapes, combined with competing attractive and repulsive interparticle interactions, lead to controllable self-assembly of structures of complex order. Applying this theory to tetrahedral NPs, we find high-yield and enantiopure self-assembly of helicoidal ribbons, exhibiting qualitative agreement with experimental observations. We expect that this theory will offer a general framework for the self-assembly of simple polyhedral building blocks into rich complex morphologies with new material capabilities such as tunable optical activity, essential for multiple emerging technologies.

Proteins and membranes form remarkably complex structures that are key to intracellular compartmentalization, cargo transport, and cell morphology. Despite this wealth of examples in living systems, we still lack design principles for controlling membrane morphology in synthetic systems. With experiments and simulations, we show that even the simple case of spherical or rod-shaped nanoparticles binding to lipid-bilayer membrane vesicles results in a remarkably rich set of morphologies that can be reliably controlled via the particle binding energy. When the binding energy is weak relative to a characteristic membrane-bending energy, vesicles adhere to one another and form a soft solid gel, which is a useful platform for controlled release. With larger binding energy, a transition from partial to complete wrapping of the nanoparticles causes a remarkable vesicle destruction process culminating in rupture, nanoparticle-membrane tubules, and vesicle inversion. We have explored the behavior across a wide range of parameter space. These findings help unify the wide range of effects observed when vesicles or cells are exposed to nanoparticles. They also show how they open the door to a new class of vesicle-based, closed-cell gels that are more than 99% water and can encapsulate and release on demand. I will discuss how triggering membrane remodeling could lead to shape-responsive systems in the future.