In search for a perfect shape of polyhedra: Buffon transformation

For an arbitrary polygon consider a new one by joining the centres of consecutive edges. Iteration of this procedure leads to a shape which is affine equivalent to a regular polygon. This regularisation effect is usually ascribed to Count Buffon (1707-1788). We discuss a natural analogue of this procedure for 3-dimensional polyhedra, which leads to a new notion of affine $B$-regular polyhedra. The main result is the proof of existence of star-shaped affine $B$-regular polyhedra with prescribed combinatorial structure, under partial symmetry and simpliciality assumptions. The proof is based on deep results from spectral graph theory due to Colin de Verdiere and Lovasz.Comment: Slightly revised version with added example of pentakis dodecahedro

RNA AS A UNIQUE POLYMER TO BUILD CONTROLLABLE NANOSTRUCTURES FOR NANOMEDICINE AND NANOTECHNOLOGY

RNA nanotechnology is an emerging field that involves the design, construction and functionalization of nanostructures composed mainly of RNA for applications in biomedical and material sciences. RNA is a unique polymer with structural simplicity like DNA and functional diversity like proteins. A variety of RNA nanostructures have been reported with different geometrical structures and functionalities. This dissertation describes the design and construction of novel two-dimensional and three-dimensional self-assembled RNA nanostructures with applications in therapeutics delivery, cancer targeting and immunomodulation. Firstly, by using the ultra-stable pRNA three-way junction motif with controllable angles and arm lengths, tetrahedral architectures composed purely of RNA were successfully assembled via one-pot bottom-up assembly with high efficiency and thermal stability. By introducing arm sizes of 22 bp and 55 bp, two RNA tetrahedrons with similar global contour structure but with different sizes of 8 nm and 17 nm were successfully assembled. The RNA tetrahedrons were also highly amenable to functionalization. Fluorogenic RNA aptamers, ribozyme, siRNA, and protein-binding RNA aptamers were integrated into the tetrahedrons by simply fusing the respective sequences with the tetrahedral core modules. Secondly, I reported the design and construction of molecularly defined RNA cages with cube and dodecahedron shapes based on the stable pRNA 3WJ. The RNA cages can be easily self-assembled by single-step annealing. The RNA cages were further characterized by gel electrophoresis, cryo-electron microscopy and atomic force microscopy, confirming the spontaneous formation of the RNA cages. I also demonstrated that the constructed RNA cages could be used to deliver model drugs such as immunomodulatory CpG DNA into cells and elicit enhanced immune responses. Thirdly, by using the modular multi-domain strategy, molecular defined RNA nanowires can be successfully self-assembled via a bottom-up approach. Only four different 44-nucleotide single-stranded RNAs were used to assemble the RNA nanowire. The reported RNA nanowire has the potential to be explored in the future as the carrier for drug delivery or matrix for tissue engineering. Fourthly, the construction of RNA polygons for delivering immunoactive CpG oligonucleotides will be presented. When CpG oligonucleotides were incorporated into the RNA polygons, their immunomodulation effect for cytokine TNF-α and IL-6 induction was greatly enhanced, while RNA polygon controls induced unnoticeable cytokine induction. Moreover, the RNA polygons were delivered to macrophages specifically and the degree of immunostimulation greatly depended on the size, shape, and the number of payload per RNA polygon. Collectively, these findings demonstrated RNA nanotechnology can produce controllable nanostructures with different functionalities and result in potential applications in nanomedicine and nanotechnology