Computational Design and Analysis of a Magic Snake

Abstract 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.

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Latent 3D Volume for Joint Depth Estimation and Semantic Segmentation from a Single Image

Sensors ◽

10.3390/s20205765 ◽

2020 ◽

Vol 20 (20) ◽

pp. 5765 ◽

Cited By ~ 1

Author(s):

Seiya Ito ◽

Naoshi Kaneko ◽

Kazuhiko Sumi

Keyword(s):

3D Structure ◽

Three Dimensional ◽

Semantic Segmentation ◽

Depth Estimation ◽

Image Features ◽

Feature Representation ◽

Single Image ◽

Feature Vectors ◽

3D Space ◽

3D Volume

This paper proposes a novel 3D representation, namely, a latent 3D volume, for joint depth estimation and semantic segmentation. Most previous studies encoded an input scene (typically given as a 2D image) into a set of feature vectors arranged over a 2D plane. However, considering the real world is three-dimensional, this 2D arrangement reduces one dimension and may limit the capacity of feature representation. In contrast, we examine the idea of arranging the feature vectors in 3D space rather than in a 2D plane. We refer to this 3D volumetric arrangement as a latent 3D volume. We will show that the latent 3D volume is beneficial to the tasks of depth estimation and semantic segmentation because these tasks require an understanding of the 3D structure of the scene. Our network first constructs an initial 3D volume using image features and then generates latent 3D volume by passing the initial 3D volume through several 3D convolutional layers. We apply depth regression and semantic segmentation by projecting the latent 3D volume onto a 2D plane. The evaluation results show that our method outperforms previous approaches on the NYU Depth v2 dataset.

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Tapered elasticæ as a route for axisymmetric morphing structures

Soft Matter ◽

10.1039/d0sm00714e ◽

2020 ◽

Vol 16 (33) ◽

pp. 7739-7750

Author(s):

Mingchao Liu ◽

Lucie Domino ◽

Dominic Vella

Keyword(s):

3D Structure ◽

Three Dimensional ◽

Two Dimensional ◽

3D Structures

Transforming flat two-dimensional (2D) sheets into three-dimensional (3D) structures by a combination of careful cutting and applied loads is an emerging manufacturing paradigm; we study how to design the cut pattern to obtain a desired 3D structure.

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Contrastive learning on protein embeddings enlightens midnight zone at lightning speed

10.1101/2021.11.14.468528 ◽

2021 ◽

Author(s):

Michael Heinzinger ◽

Maria Littmann ◽

Ian Sillitoe ◽

Nicola Bordin ◽

Christine Orengo ◽

...

Keyword(s):

Structure Prediction ◽

Sequence Similarity ◽

3D Structure ◽

Three Dimensional ◽

Hierarchical Classification ◽

Language Models ◽

Sequence Alignments ◽

Sequence Comparisons ◽

Multiple Sequence ◽

3D Structures

Thanks to the recent advances in protein three-dimensional (3D) structure prediction, in particular through AlphaFold 2 and RoseTTAFold, the abundance of protein 3D information will explode over the next year(s). Expert resources based on 3D structures such as SCOP and CATH have been organizing the complex sequence-structure-function relations into a hierarchical classification schema. Experimental structures are leveraged through multiple sequence alignments, or more generally through homology-based inference (HBI) transferring annotations from a protein with experimentally known annotation to a query without annotation. Here, we presented a novel approach that expands the concept of HBI from a low-dimensional sequence-distance lookup to the level of a high-dimensional embedding-based annotation transfer (EAT). Secondly, we introduced a novel solution using single protein sequence representations from protein Language Models (pLMs), so called embeddings (Prose, ESM-1b, ProtBERT, and ProtT5), as input to contrastive learning, by which a new set of embeddings was created that optimized constraints captured by hierarchical classifications of protein 3D structures. These new embeddings (dubbed ProtTucker) clearly improved what was historically referred to as threading or fold recognition. Thereby, the new embeddings enabled the intrusion into the midnight zone of protein comparisons, i.e., the region in which the level of pairwise sequence similarity is akin of random relations and therefore is hard to navigate by HBI methods. Cautious benchmarking showed that ProtTucker reached much further than advanced sequence comparisons without the need to compute alignments allowing it to be orders of magnitude faster. Code is available at https://github.com/Rostlab/EAT .

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A tool for multi-scale modelling of the renal nephron

Interface Focus ◽

10.1098/rsfs.2010.0032 ◽

2011 ◽

Vol 1 (3) ◽

pp. 417-425 ◽

Cited By ~ 7

Author(s):

David P. Nickerson ◽

Jonna R. Terkildsen ◽

Kirk L. Hamilton ◽

Peter J. Hunter

Keyword(s):

Mathematical Model ◽

Spatial Scales ◽

Three Dimensional ◽

Scale Model ◽

Model Description ◽

Dimensional Model ◽

Three Dimensional Model ◽

One Dimensional ◽

Multi Scale ◽

The Mathematical Model

We present the development of a tool, which provides users with the ability to visualize and interact with a comprehensive description of a multi-scale model of the renal nephron. A one-dimensional anatomical model of the nephron has been created and is used for visualization and modelling of tubule transport in various nephron anatomical segments. Mathematical models of nephron segments are embedded in the one-dimensional model. At the cellular level, these segment models use models encoded in CellML to describe cellular and subcellular transport kinetics. A web-based presentation environment has been developed that allows the user to visualize and navigate through the multi-scale nephron model, including simulation results, at the different spatial scales encompassed by the model description. The Zinc extension to Firefox is used to provide an interactive three-dimensional view of the tubule model and the native Firefox rendering of scalable vector graphics is used to present schematic diagrams for cellular and subcellular scale models. The model viewer is embedded in a web page that dynamically presents content based on user input. For example, when viewing the whole nephron model, the user might be presented with information on the various embedded segment models as they select them in the three-dimensional model view. Alternatively, the user chooses to focus the model viewer on a cellular model located in a particular nephron segment in order to view the various membrane transport proteins. Selecting a specific protein may then present the user with a description of the mathematical model governing the behaviour of that protein—including the mathematical model itself and various simulation experiments used to validate the model against the literature.

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ASSESSING THE QUALITY OF THE HOMOLOGY-MODELED 3D STRUCTURES FROM ELECTROSTATIC STANDPOINT: TEST ON BACTERIAL NUCLEOSIDE MONOPHOSPHATE KINASE FAMILIES

Journal of Bioinformatics and Computational Biology ◽

10.1142/s0219720007002709 ◽

2007 ◽

Vol 05 (03) ◽

pp. 693-715 ◽

Cited By ~ 3

Author(s):

PETRAS KUNDROTAS ◽

PAULINA GEORGIEVA ◽

ALEXANDRA SHOSHEVA ◽

PETYA CHRISTOVA ◽

EMIL ALEXOV

Keyword(s):

3D Structure ◽

Three Dimensional ◽

3D Models ◽

Structural Defects ◽

Large Protein ◽

Homologous Proteins ◽

X Ray ◽

3D Structures ◽

Large Sets

In this study, we address the issue of performing meaningful pKa calculations using homology modeled three-dimensional (3D) structures and analyze the possibility of using the calculated pKa values to detect structural defects in the models. For this purpose, the 3D structure of each member of five large protein families of a bacterial nucleoside monophosphate kinases (NMPK) have been modeled by means of homology-based approach. Further, we performed pKa calculations for the each model and for the template X-ray structures. Each bacterial NMPK family used in the study comprised on average 100 members providing a pool of sequences and 3D models large enough for reliable statistical analysis. It was shown that pKa values of titratable groups, which are highly conserved within a family, tend to be conserved among the models too. We demonstrated that homology modeled structures with sequence identity larger than 35% and gap percentile smaller than 10% can be used for meaningful pKa calculations. In addition, it was found that some highly conserved titratable groups either exhibit large pKa fluctuations among the models or have pKa values shifted by several pH units with respect to the pKa calculated for the X-ray structure. We demonstrated that such case usually indicates structural errors associated with the model. Thus, we argue that pKa calculations can be used for assessing the quality of the 3D models by monitoring fluctuations of the pKa values for highly conserved titratable residues within large sets of homologous proteins.

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RNA 3D Structure Prediction Using Coarse-Grained Models

Frontiers in Molecular Biosciences ◽

10.3389/fmolb.2021.720937 ◽

2021 ◽

Vol 8 ◽

Author(s):

Jun Li ◽

Shi-Jie Chen

Keyword(s):

Structure Prediction ◽

3D Structure ◽

Three Dimensional ◽

Coarse Grained ◽

Biological Functions ◽

3D Structures ◽

Rna Molecules ◽

Atomic Structures ◽

Long Time

The three-dimensional (3D) structures of Ribonucleic acid (RNA) molecules are essential to understanding their various and important biological functions. However, experimental determination of the atomic structures is laborious and technically difficult. The large gap between the number of sequences and the experimentally determined structures enables the thriving development of computational approaches to modeling RNAs. However, computational methods based on all-atom simulations are intractable for large RNA systems, which demand long time simulations. Facing such a challenge, many coarse-grained (CG) models have been developed. Here, we provide a review of CG models for modeling RNA 3D structures, compare the performance of the different models, and offer insights into potential future developments.

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NONLINEAR MODEL OF A FABRIC WARP AND WEFT

Advances in Complex Systems ◽

10.1142/s0219525906000707 ◽

2006 ◽

Vol 09 (01n02) ◽

pp. 99-120 ◽

Cited By ~ 3

Author(s):

PASCAL BRUNIAUX ◽

CYRIL NGO NGOC

Keyword(s):

Dynamic Behavior ◽

3D Structure ◽

Three Dimensional ◽

Geometrical Model ◽

Nonlinear Phenomena ◽

Initial State ◽

Numerical Resolution ◽

3D Space ◽

Proposed Model ◽

Nodal Model

This study aims to develop a realistic mathematical model of fabric. In contrast to other studies on fabric modeling as a deformable surface, the model described in this article takes into account the geometry of the object. Moreover, it integrates the nonlinear phenomena of the dynamic behavior of material. As input parameters, the weaving data that define the 3D structure of the object and the mechanical properties of the yarn that express its dynamics are used. Thus, the fabric model is composed of a geometrical model of fabric (structure) on which a model of yarn (material characterization) is added. This hypothesis may be reasonable since a fabric shows the result of a three-dimensional assembly of yarns judiciously disposed. Since these yarns interact dynamically: the main difficulty consists of defining the yarn model. In our case, it is composed of various nonlinear functions representing the dynamic behavior of yarn. In order to characterize the flexibility of material, the weight, the elasticity and any other mechanical characteristics defining the relation between the strain and the stretching out of the shape should be taken into account. Firstly, several works dealing with realistic mathematical models of fabric are described. A taxonomic classification is achieved in order to position our study (in comparison to the scientific community). Secondly, the model of the fabric is described. A geometrical model of the object is presented. It allows one to dimension the object in a 3D space and then to position it at its initial state. Subsequently, a nodal model of yarns is described, step by step, in order to demonstrate the separability of the various dynamic behaviors. These nodal links make it simple to integrate the proposed model in the global geometrical model. Thus, the methods of numerical resolution used to simulate the complete model of the fabric are exposed. One method is selected and used in order to improve the performances of the fabric simulator and to obtain better stability. Several simulations illustrate the quality of the results obtained.

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Three-Dimensional Representation of Curved Nanostructures

MRS Proceedings ◽

10.1557/proc-820-r8.4 ◽

2004 ◽

Vol 820 ◽

Author(s):

Z. Huang ◽

D.A. Dikin ◽

W. Ding ◽

Y. Qiao ◽

Y. Fridman ◽

...

Keyword(s):

Three Dimensional ◽

Accurate Estimate ◽

One Dimensional ◽

3D Space ◽

Precise Knowledge ◽

Three Dimensional Representation ◽

Resonance Data ◽

Image Pairs ◽

Program Error ◽

Boron Nanowires

AbstractNanostructures, such as nanowires, nanotubes, and nanocoils, can be described in many cases as quasi one-dimensional (1D) curved objects projecting in three-dimensional (3D) space. A parallax method to reconstruct the correct three-dimensional geometry of such 1D nanostructures is presented. A series of images were acquired at different view angles, and from those image pairs, 3D representations were constructed using a MATLAB program. Error analysis as a function of view-angle between the two images is discussed. As an example application, we demonstrate the importance of knowing the true 3D shape of Boron nanowires. Without precise knowledge of the nanowire's dimensions, diameter and length, mechanical resonance data cannot be properly fit to obtain an accurate estimate of the Young's modulus.

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A Heterometallic Three-Dimensional Metal−Organic Framework Bearing an Unprecedented One-Dimensional Branched-Chain Secondary Building Unit

Molecules ◽

10.3390/molecules25092190 ◽

2020 ◽

Vol 25 (9) ◽

pp. 2190

Author(s):

Jing Chen ◽

Meng-Yao Chao ◽

Yan Liu ◽

Bo-Wei Xu ◽

Wen-Hua Zhang ◽

...

Keyword(s):

Metal Organic Framework ◽

3D Structure ◽

Three Dimensional ◽

Solvothermal Reaction ◽

Secondary Building Unit ◽

One Dimensional ◽

Building Unit ◽

Metal Organic ◽

Branched Chain ◽

Organic Framework

A heterometallic metal−organic framework (MOF) of [Cd6Ca4(BTB)6(HCOO)2(DEF)2(H2O)12]∙DEF∙xSol (1, H3BTB = benzene-1,3,5-tribenzoic acid; DEF = N,N′-diethylformamide; xSol. = undefined solvates within the pore) was prepared by solvothermal reaction of Cd(NO3)2·4H2O, CaO and H3BTB in a mixed solvent of DEF/H2O/HNO3. The compatibility of these two divalent cations from different blocks of the periodic table results in a solid-state structure consisting of an unusual combination of a discrete V-shaped heptanuclear cluster of [Cd2Ca]2Ca′ and an infinite one-dimensional (1D) chain of [Cd2CaCa′]n that are orthogonally linked via a corner-shared Ca2+ ion (denoted as Ca′), giving rise to an unprecedented branched-chain secondary building unit (SBU). These SBUs propagate via tridentate BTB to yield a three-dimensional (3D) structure featuring a corner-truncated P41 helix in MOF 1. This outcome highlights the unique topologies possible via the combination of carefully chosen s- and d-block metal ions with polydentate ligands.

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P-Spice Modeling Three-Dimensional Propagation of Ultrasound Diffraction Considering a Gaussian Beam Approach

Journal of Vibration and Acoustics ◽

10.1115/1.4031824 ◽

2015 ◽

Vol 138 (1) ◽

Cited By ~ 1

Author(s):

Soucrati Hanane ◽

Chitnalah Ahmed ◽

Aouzale Noureddine ◽

El Idrissi Abdelaziz

Keyword(s):

Wave Propagation ◽

Three Dimensional ◽

Gaussian Beams ◽

Average Pressure ◽

Transmission Line Model ◽

Spice Simulation ◽

Ultrasonic Wave Propagation ◽

One Dimensional ◽

Spice Model ◽

3D Space

In this paper, we propose a new method for simulating three-dimensional (3D) ultrasonic wave propagation using P-Spice like simulator. We use a one-dimensional transmission line model to implement the diffraction losses. In order to simulate the beam pattern considering axial and radial orientations, we calculate the diffraction losses in 3D space. First, we express the radiated field using a set of Gaussian beams. Calculating the average pressure over the receiver surface allows us to determine the diffraction losses. These losses are then incorporated into the P-Spice model via the G parameter which is axial and radial orientations dependent. Comparison between P-Spice simulation and analytical model results shows good agreements.

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