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The auxetic structures considered in this paper are three-dimensional periodic bar-and-joint frameworks. We start with the specific purpose of obtaining an auxetic design with underlying periodic graph of low valency. Adapting a general methodology, we produce an initial framework with valency seven and one degree of freedom. Then, we describe a saturation process, whereby edge orbits are added up to valency 16, with no alteration of the deformation path. This is reflected in a large dimension for the space of periodic self-stresses. The saturated version has higher crystallographic symmetry and allows a precise description of the deformation trajectory. Reducing saturation by adequate removal of edge orbits results in vast numbers of distinct auxetic designs which obey the same kinematics., (© 2022 The Authors.)

In materials science and engineering, auxetic behavior refers to deformations of flexible structures where stretching in some direction involves lateral widening, rather than lateral shrinking. We address the problem of detecting auxetic behavior for flexible periodic bar-and-joint frameworks. Currently, the only known algorithmic solution is based on the rather heavy machinery of fixed-dimension semi-definite programming. In this paper we present a new, simpler algorithmic approach which is applicable to a natural family of three-dimensional periodic bar-and-joint frameworks with three degrees of freedom. This class includes most zeolite structures, which are important for applications in computational materials science. We show that the existence of auxetic deformations is related to properties of an associated elliptic curve. A fast algorithm for recognizing auxetic capabilities is obtained via the classical Aronhold invariants of the cubic form defining the curve. A related alternative is also considered.

KINematics And RIgidity (KINARI) is an on-going project for in silico flexibility analysis of proteins. The new version of the software, Kinari-2, extends the functionality of our free web server KinariWeb, incorporates advanced web technologies, emphasizes the reproducibility of its experiments, and makes substantially improved tools available to the user. It is designed specifically for large scale experiments, in particular, for (a) very large molecules, including bioassemblies with high degree of symmetry such as viruses and crystals, (b) large collections of related biomolecules, such as those obtained through simulated dilutions, mutations, or conformational changes from various types of dynamics simulations, and (c) is intended to work as seemlessly as possible on the large, idiosyncratic, publicly available repository of biomolecules, the Protein Data Bank. We describe the system design, along with the main data processing, computational, mathematical, and validation challenges underlying this phase of the KINARI project.

Background: We initiate in silico rigidity-theoretical studies of biological assemblies and small crystals for protein structures. The goal is to determine if, and how, the interactions among neighboring cells and subchains affect the flexibility of a molecule in its crystallized state. We use experimental X-ray crystallography data from the Protein Data Bank (PDB). The analysis relies on an effcient graph-based algorithm. Computational experiments were performed using new protein rigidity analysis tools available in the new release of our KINARI-Web server http://kinari.cs.umass.edu., Results: We provide two types of results: on biological assemblies and on crystals. We found that when only isolated subchains are considered, structural and functional information may be missed. Indeed, the rigidity of biological assemblies is sometimes dependent on the count and placement of hydrogen bonds and other interactions among the individual subchains of the biological unit. Similarly, the rigidity of small crystals may be affected by the interactions between atoms belonging to different unit cells., Conclusion: The rigidity analysis of a single asymmetric unit may not accurately reflect the protein's behavior in the tightly packed crystal environment. Using our KINARI software, we demonstrated that additional functional and rigidity information can be gained by analyzing a protein's biological assembly and/or crystal structure. However, performing a larger scale study would be computationally expensive (due to the size of the molecules involved). Overcoming this limitation will require novel mathematical and computational extensions to our software.

Background: Protein rigidity analysis is an efficient computational method for extracting flexibility information from static, X-ray crystallography protein data. Atoms and bonds are modeled as a mechanical structure and analyzed with a fast graph-based algorithm, producing a decomposition of the flexible molecule into interconnected rigid clusters. The result depends critically on noncovalent atomic interactions, primarily on how hydrogen bonds and hydrophobic interactions are computed and modeled. Ongoing research points to the stringent need for benchmarking rigidity analysis software systems, towards the goal of increasing their accuracy and validating their results, either against each other and against biologically relevant (functional) parameters. We propose two new methods for modeling hydrogen bonds and hydrophobic interactions that more accurately reflect a mechanical model, without being computationally more intensive. We evaluate them using a novel scoring method, based on the B-cubed score from the information retrieval literature, which measures how well two cluster decompositions match., Results: To evaluate the modeling accuracy of KINARI, our pebble-game rigidity analysis system, we use a benchmark data set of 20 proteins, each with multiple distinct conformations deposited in the Protein Data Bank. Cluster decompositions for them were previously determined with the RigidFinder method from Gerstein's lab and validated against experimental data. When KINARI's default tuning parameters are used, an improvement of the B-cubed score over a crude baseline is observed in 30% of this data. With our new modeling options, improvements were observed in over 70% of the proteins in this data set. We investigate the sensitivity of the cluster decomposition score with case studies on pyruvate phosphate dikinase and calmodulin., Conclusion: To substantially improve the accuracy of protein rigidity analysis systems, thorough benchmarking must be performed on all current systems and future extensions. We have measured the gain in performance by comparing different modeling methods for noncovalent interactions. We showed that new criteria for modeling hydrogen bonds and hydrophobic interactions can significantly improve the results. The two new methods proposed here have been implemented and made publicly available in the current version of KINARI (v1.3), together with the benchmarking tools, which can be downloaded from our software's website, http://kinari.cs.umass.edu.