We proved several theorems about the Rubik's snake toy, which has a chain-like structure with applications in robot design. These mathematical theories can guide people to design beautiful shapes using the Rubik's snake. We presented an example for shape design to demonstrate the effectiveness of our approach.

The Magic Snake (Rubik’s Snake) is a toy that was invented decades ago. It draws much less attention than Rubik’s Cube, which was invented by the same professor, Erno Rubik. The number of configurations of a Magic Snake, determined by the number of discrete rotations about the elementary wedges in a typical snake, is far less than the possible configurations of a typical cube. However, a cube has only a single three-dimensional (3D) structure while the number of sterically allowed 3D conformations of the snake is unknown. Here, we demonstrate how to represent a Magic Snake as a one-dimensional (1D) sequence that can be converted into a 3D structure. We then provide two strategies for designing Magic Snakes to have specified 3D structures. The first enables the folding of a Magic Snake onto any 3D space curve. The second introduces the idea of “embedding” to expand an existing Magic Snake into a longer, more complex, self-similar Magic Snake. Collectively, these ideas allow us to rapidly list and then compute all possible 3D conformations of a Magic Snake. They also form the basis for multidimensional, multi-scale representations of chain-like structures and other slender bodies including certain types of robots, polymers, proteins, and DNA.

Nucleosomes are the fundamental building blocks of chromatin, the biomaterial that houses the genome in all higher organisms. A nucleosome consists of 145−147 base pairs of DNA wrapped 1.7 times around eight histones. Given a four-letter code (A, C, G, T), there are approximately 4 147 or 10 88 oligonucleotides that can form a nucleosome. Comparative, rather than comprehensive, studies are required. Here we introduce the TMB Library of nucleosome simulations and present a metaanalysis of over 20 μs of all atom molecular dynamics simulations representing 518 different realizations of the nucleosome. The TMB Library serves as a reference for future comparative, on-demand simulations of nucleosomes and a demonstration of iBIOMES Lite as a tool for managing a laboratory's simulation library. For every simulation, dewatered trajectories, RMSD, and DNA helical parameter data are provided through iBIOMES Lite in a Web browser and a file browser format. A novel view of nucleosomal DNA emerges from our meta-analysis of the TMB Library. DNA conformation is restricted to a specific left-handed superhelix, but the range of conformations observed for individual bases and base pairs is not more restricted nor more highly deformed than DNA free in solution. With the exception of Roll, mean DNA helical parameter values obtained from simulations of nucleosomes are largely within the range of thermal motion of DNA free in solution. The library provides evidence of DNA kinking in the nucleosome and clearly demonstrates the effects of DNA sequence on the gross structure and dynamics of nucleosomes. These effects and mispositioning of the 601 super strong nucleosome positioning sequence can be detected in short simulations (10 ns). Collectively, the results provide a basis for comparative simulation studies of nucleosomes and extend our understanding of the binding of proteins and drugs to nucleosomal DNA. The TMB Library can be found at http://dna.engr.latech.edu/~tmbshare/.

Genomics is a sequence based informatics science and a structure based molecular material science. There are few tools available that unite these approaches in a scientifically robust manner. Here we describe G-Dash, a web based prototype of a genomics dashboard, specifically designed to integrate informatics and 3D material studies of chromatin. G-Dash unites our Interactive Chromatin Modeling(ICM) tools with the Biodalliance genome browser and the JSMol molecular viewer to rapidly fold any DNA sequence into atomic or coarse-grained models of DNA, nucleosomes or chromatin. As a chromatin modeling tool, G-Dash enables users to specify nucleosome positions from various experimental or theoretical sources, interactively manipulate nucleosomes, and assign different conformational states to each nucleosome. As an informatics tool, data associated with 3D structures are displayed as tracks in a genome browser. The exchange of data between informatics and structure is bi-directional so any informatics track can inform a molecular structure (e.g. color by function) and structure features can be displayed as informatics tracks in a genome browser(e.g. Roll, Slide, or Twist). As a sample application, models of the CHA1 promoter based on experimentally determined nucleosome positions are explored with G-Dash. Steric clashes and DNA knotting are observed but can be resolved with G-Dash's minimal coarse-grained model without significant variation in structure. Results raise questions about the interpretation of nucleosome positioning data and promoter structures. In this regard, G-Dash is a novel tool for investigating structure-function relationships for regions of the genome ranging from base pairs to chromosomes and for generating, validating and testing mechanistic hypotheses.

<div>Here we introduce our TMB-iBIOMES database which serves as both a reference for future comparative, on-demand simulations of nucleosomes and a demonstration of iBIOMES-Lite as a tool for managing a laboratory's simulation inventory. TMB-iBIOMES contains over 20 microseconds of all atom molecular dynamics simulations for over 500 different realizations of the nucleosome. For every simulation, the original input, output, de-watered trajectories, RMSD, and DNA helical parameter data are provided. Closely related simulations are grouped together, and a meta-analysis of each group is provided. The data can be navigated in a file browser format or downloaded directly with command line tools. Collectively the simulations provide a novel view of nucleosomal DNA. Compared to DNA free in solution, DNA on the nucleosome is not highly deformed or tightly restricted as determined by DNA helical parameter analysis. The overall conformation is restricted to a specific left-handed super helix, but the range of conformations explored by individual base pairs is larger than that observed for DNA free in solution. </div>