Duotone Surfaces: Division of a Closed Surface into Exactly Two Regions

In this thesis work, our main motivation is to create computer aided art work which can eventually transform into a sculpting tool. The work was inspired after Taubin’s work on constructing Hamiltonian triangle strips on quadrilateral meshes. We present an algorithm that can divide a closed 2-manifold surface into exactly two regions, bounded from each other by a single continuous curve. We show that this kind of surface division is possible only if the mesh approximation of a given object is a two colorable quadrilateral mesh. For such a quadrilateral mesh, appropriate texturing of the faces of the mesh using a pair of Truchet tiles will give us a Duotone Surface. Catmull-Clark subdivision can convert any given mesh with arbitrary sided polygons into a two colorable quadrilateral mesh. Using such vertex insertion schemes, we modify the mesh and classify the vertices of the new mesh into two sets. By appropriately texturing each face of the mesh such that the color of the vertices of the face match with the colored regions of the corresponding Truchet tile, we can get a continuous curve that splits the surface of the mesh into two regions. Now, coloring the thus obtained two regions with two different colors gives us a Duotone Surface

Programmable disorder in random DNA tilings

Scaling up the complexity and diversity of synthetic molecular structures will require strategies that exploit the inherent stochasticity of molecular systems in a controlled fashion. Here we demonstrate a framework for programming random DNA tilings and show how to control the properties of global patterns through simple, local rules. We constructed three general forms of planar network—random loops, mazes and trees—on the surface of self-assembled DNA origami arrays on the micrometre scale with nanometre resolution. Using simple molecular building blocks and robust experimental conditions, we demonstrate control of a wide range of properties of the random networks, including the branching rules, the growth directions, the proximity between adjacent networks and the size distribution. Much as combinatorial approaches for generating random one-dimensional chains of polymers have been used to revolutionize chemical synthesis and the selection of functional nucleic acids, our strategy extends these principles to random two-dimensional networks of molecules and creates new opportunities for fabricating more complex molecular devices that are organized by DNA nanostructures

Programmable disorder in random DNA tilings

Scaling up the complexity and diversity of synthetic molecular structures will require strategies that exploit the inherent stochasticity of molecular systems in a controlled fashion. Here we demonstrate a framework for programming random DNA tilings and show how to control the properties of global patterns through simple, local rules. We constructed three general forms of planar network—random loops, mazes and trees—on the surface of self-assembled DNA origami arrays on the micrometre scale with nanometre resolution. Using simple molecular building blocks and robust experimental conditions, we demonstrate control of a wide range of properties of the random networks, including the branching rules, the growth directions, the proximity between adjacent networks and the size distribution. Much as combinatorial approaches for generating random one-dimensional chains of polymers have been used to revolutionize chemical synthesis and the selection of functional nucleic acids, our strategy extends these principles to random two-dimensional networks of molecules and creates new opportunities for fabricating more complex molecular devices that are organized by DNA nanostructures

Engineering Molecular Self-assembly and Reconfiguration in DNA Nanostructures

Smart electronics have developed ubiquitously to assist people in everything from navigation to health monitoring. The rise of complex electronics relied on rational design of platforms to build ever larger and more complex circuit networks and for frameworks to test those electronics. Biochemical circuits have also seen dramatic advancement in the last two decades within the field of DNA nanotechnology. As with electronics, DNA nanotechnology applied rational design to DNA molecules to build ever more complex biochemical networks that, beyond current electronics, also retain a significant measure of biological compatibility and plasticity akin to many networks of biological origin. Well situated for promising applications in diagnostics and therapeutics, advancing DNA nanotechnology devices will also rely upon larger platforms and testing frameworks. In roughly the last decade, researchers have been building upon the invention of DNA origami, a technique allowing the robust construction of biomolecular nano-structures capable of precise nanometer positioning of proteins, nanoparticles, and other molecules. DNA circuits have computed on the nanostructures; DNA robots have moved nanoparticles, made choices, and have even sorted cargo on the surface of a nanostructure. The complexity of circuits and devices continues to rise. In this thesis, we will discuss our contributions to the field of DNA nanotechnology by developing design rules and systematic approaches to controlling nanostructure complex assembly. These rules and approaches allow for the construction of molecular structures with a tunable diversity, large systems approaching the size of bacteria yet retaining nanometer precision, and biological plasticity inspired dynamic systems for arbitrary reconfiguration. Using a DNA origami tile tailored for array formation with a high continuous surface area, we create a framework inspired from molecular stochasticity for programming DNA array formation and gaining control over diversity of global properties through simple local rules. Three general forms of planar networks, random loops, mazes, and trees, were manipulated on the micron scale upon the self-assembled DNA arrays. We demonstrate control of several properties of the networks, such as branching rules, growth directions, the proximity between adjacent networks, and size distributions. The large diversity, in principle, allows for a wide, but tunable, testing environment for molecular circuits. By further applying these principles to subunits of finite assemblies, variable components may be mixed with fixed components potentially opening additional applications in high throughput device or drug screening. Next we turned to expanding the platform size biochemical circuits may be built upon. While DNA origami allows nanometer precise placement, the size remains roughly below 0.05 um2. Toward making large arbitrarily complex structures with only a set of simple tiles, multi-stage self-assembly has been explored in theory and for small DNA tiles. None were successful experimentally with DNA origami. We developed a strategy for DNA origami: a simple rule set applied recursively in each stage of a hierarchical self-assembly process, and to significantly reduce costs, a constant set of unique DNA strands regardless of size. We also developed a software tool to automatically compile a designed surface pattern into experimental protocols. We experimentally demonstrated DNA origami arrays approaching the size of small bacteria, 0.5 um2, with several arbitrary patterns, each consisting of 8,704 specifically chosen pixel locations with nanometer precision, including a bacteria sized portrait of a bacteria. The large platform opens the door to more advanced molecular circuits for applications such as diagnostics. Finally we demonstrated control over the dynamics of DNA origami reconfiguration in tile arrays. In an approach we call DNA tile displacement, we showed that a DNA origami array may have tiles arbitrarily replaced by another tile, including tiles of another shape or surface pattern. We also demonstrated control over the kinetics of tile displacement and performed several general purpose reconfigurations of DNA nanostructures. Examples include sequential reconfiguration, competitive reconfiguration, cooperative reconfiguration, and finally the scalability of multi-step reconfiguration as demonstrated through a fully playable nano-scale biomolecular tic-tac-toe game. The major ramifications are a plasticity more common to biology than to electronics—molecular platforms with arbitrary patterning that can reconfigure an arbitrary part of the nanostructure in an arbitrary order based on environmental signals. In principle, such reconfiguration can allow advanced circuits with the capacity to adapt to environmental needs or heal damaged components.</p