Rigidity Theory and Applications

We show that the negative of the number of floppy modes behaves as a free energy for both connectivity and rigidity percolation, and we illustrate this result using Bethe lattices. The rigidity transition on Bethe lattices is found to be first order at a bond concentration close to that predicted by Maxwell constraint counting. We calculate the probability of a bond being on the infinite cluster and also on the overconstrained part of the infinite cluster, and show how a specific heatcan be defined as the second derivative of the free energy. We demonstrate that the Bethe lattice solution is equivalent to that of the random bond model, where points are joined randomly (with equal probability at all length scales) to have a given coordination, and then subsequently bonds are randomly removed.

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Bulk and surface floppy modes

Using a recently developed algorithm for generic rigidity of two-dimensional graphs, we analyze rigidity and connectivity percolation transitions in two dimensions on lattices of linear size up to $L=4096.$ We compare three different universality classes: the generic rigidity class, the connectivity class, and the generic ``braced square net''(GBSN). We analyze the spanning cluster density ${P}_{\ensuremath{\infty}},$ the backbone density ${P}_{B}$, and the density of dangling ends ${P}_{D}.$ In the generic rigidity (GR) and connectivity cases, the load-carrying component of the spanning cluster, the backbone, is fractal at ${p}_{c},$ so that the backbone density behaves as $B\ensuremath{\sim}(p\ensuremath{-}{p}_{c}{)}^{{\ensuremath{\beta}}^{\ensuremath{'}}}$ for $p>{p}_{c}.$ We estimate ${\ensuremath{\beta}}_{\mathrm{gr}}^{\ensuremath{'}}=0.25\ifmmode\pm\else\textpm\fi{}0.02$ for generic rigidity and ${\ensuremath{\beta}}_{c}^{\ensuremath{'}}=0.467\ifmmode\pm\else\textpm\fi{}0.007$ for the connectivity case. We find the correlation length exponents ${\ensuremath{\nu}}_{\mathrm{gr}}=1.16\ifmmode\pm\else\textpm\fi{}0.03$ for generic rigidity compared to the exact value for connectivity, ${\ensuremath{\nu}}_{c}=\frac{4}{3}.$ In contrast the GBSN undergoes a first-order rigidity transition, with the backbone density being extensive at ${p}_{c},$ and undergoing a jump discontinuity on reducing p across the transition. We define a model which tunes continuously between the GBSN and GR classes, and show that the GR class is typical.

An Algorithm for Two-Dimensional Rigidity Percolation: The Pebble Game

Topological constraint theory (TCT) has revealed itself to be a powerful tool in interpreting the behaviors of amorphous solids. The theory predicts a transition between a “rigid” overconstrained network and a “floppy” underconstrained network as a function of connectivity or average coordination number, 〈r〉. The predicted results have been shown experimentally for various glassy materials, the majority of these being based on 4-fold-coordinate networks such as chalcogenide and oxide glasses. Here, we demonstrate the broader applicability of topological constraint theory to uniquely coordinated amorphous hydrogenated boron carbide (a-BC:H), based on 6-fold-coordinate boron atoms arranged into partially hydrogenated interconnected 12-vertex icosahedra. We have produced a substantial set of plasma-enhanced chemical vapor deposited a-BC:H films with a large range of densities and network coordination, and demonstrate a clear threshold in Young's modulus as a function of 〈r〉, ascribed to a rigidity transition. We investigate constraint counting strategies in this material and show that by treating icosahedra as “superatoms,” a rigidity transition is observed within the range of the theoretically predicted 〈r〉c value of 2.4 for covalent solids with bond-stretching and bond-bending forces. This experimental data set for a-BC:H is unique in that it represents a uniform change in connectivity with 〈r〉 and demonstrates a distinct rigidity transition with data points both above and below the transition threshold. Finally, we discuss how TCT can be applied to explain and optimize mechanical and dielectric properties in a-BC:H and related materials in the context of microelectronics applications.

Advancements in the field of mathematical rigidity theory have opened up a number of exciting opportunities for computational predictions of protein flexibility and their dynamics. Starting with a 3D protein structure, several programs such as FIRST model the protein as a constraint multigraph, consisting of vertices (atoms) and edges (covalent bonds, hydrogen bonds, electrostatic interactions, and hydrophobic contacts). FIRST applies the pebble game algorithm on the resulting multigraph which rapidly decompose the protein into rigid clusters and flexible regions. Using an extension of FIRST and the pebble game algorithm we propose a computational approach for studying a biological phenomenon 'allostery'. Allostery refers to an effect of binding at one site to another, often significantly distant functional site on the protein, allowing for regulation of the protein function. Most dynamic proteins are allosteric and allostery has even been coined the 'second secret of life', however the molecular mechanisms that give rise to allostery are currently poorly understood. Extending our earlier seminal work, we have developed a rigidity-transmission allostery (RTA) algorithm which predicts if mechanical perturbation of rigidity (mimicking ligand binding) at one site of the protein can propagate across a protein structure and in turn cause a transmission and change in degrees of freedom and conformation at a second distant site, resulting in allosteric transmission. Since RTA algorithm is computationally fast, we can rapidly scan many unknown sites for rigidity-based allosteric communication, identifying potential new allosteric sites and quantify their allosteric effect. We will review the functional importance of protein flexibility and mathematical and algorithmic background of rigidity theory and method FIRST. In this originative expose we describe rigidity based mechanistic allostery communication model. We will also provide a few illustrations of rigidity-based allostery communication on actual protein structures, including the important signaling G-protein coupled receptors. This method will have important consequence in general understanding of allostery and in aid of design of allosteric drugs.

Theoretic analysis of unique localization for wireless sensor networks

Structural disorder has a dramatic impact on the mechanical response and stability of solids. On the one hand, rigidity percolation shows that the limit of mechanical stability coincides with the emergence of floppy modes. On the other hand, the rigidity of solids is also lowered by nonaffine atomic displacements, i.e. additional motions caused by the disorder on top of the affine displacements dictated by the strain. These two frameworks have offered alternative descriptions of the elasticity of disordered solids with central-force bonds, but the relationship between rigidity percolation and nonaffinity has remained unclear. As such, a unifying theory of real materials, i.e. those with covalent (noncentral) bonds, such as amorphous semiconductors, has been elusive. After briefly reviewing these theories, we present a mean-field argument which attempts to provide the unifying link between rigidity percolation and non-affinity. This framework yields analytical predictions of the shear modulus of covalent amorphous solids with potential applications to amorphous semiconductors and disordered carbon electronic materials.

A fast computer algorithm, the pebble game, has been used successfully to analyze the rigidity of two-dimensional (2D) elastic networks, as well as of a special class of 3D networks, the bond-bending networks, and enabled significant progress in studies of rigidity percolation on such networks. Application of the pebble game approach to general 3D networks has been hindered by the fact that the underlying mathematical theory is, strictly speaking, invalid in this case. We construct an approximate pebble game algorithm for general 3D networks, as well as a slower but exact algorithm, the relaxation algorithm, that we use for testing the new pebble game. Based on the results of these tests and additional considerations, we argue that in the particular case of randomly diluted central-force networks on bcc and fcc lattices, the pebble game is essentially exact. Using the pebble game, we observe an extremely sharp jump in the largest rigid cluster size in bond-diluted central-force networks in 3D, with the percolating cluster appearing and taking up most of the network after a single bond addition. This strongly suggests a first-order rigidity percolation transition, which is in contrast to the second-order transitions found previously for the 2D central-force and 3D bond-bending networks. While a first order rigidity transition has been observed previously for Bethe lattices and networks with "chemical order," here it is in a regular randomly diluted network. In the case of site dilution, the transition is also first order for bcc lattices, but results for fcc lattices suggest a second-order transition. Even in bond-diluted lattices, while the transition appears massively first order in the order parameter (the percolating cluster size), it is continuous in the elastic moduli. This, and the apparent nonuniversality, make this phase transition highly unusual.

Pseudomycin A is a cyclic lipodepsinonapeptide phytotoxin produced by a strain of the plant pathogenic bacterium Pseudomonas syringae. Like other members of this family of bacterial metabolites, it is characterised by a fatty acylated cyclic peptide with mixed chirality and lactonic closure. Several biological activities of Pseudomycin A are lower than those found for some of its congeners, a difference which might depend on the diverse number and distribution of charged residues in the peptide moiety. Hence, it was of interest to investigate its conformation in solution. After the complete interpretation of the two-dimensional NMR spectra, NOE data were obtained and the structure was determined by computer simulations, applying distance geometry and molecular dynamics procedures. The conformation of the large ring of Pseudomycin A in solution includes three rigid structural regions interrupted by three short flexible regions that act as hinges. The overall three-dimensional structure of the cyclic moiety is similar to that of previously studied bioactive lipodepsinonapeptides produced by other pseudomonads.

In order to understand the mechanisms for glassy dynamics in biological tissues and shed light on those in non-biological materials, we study the low-temperature disordered phase of 2D vertex-like models. Recently it has been noted that vertex models have quite unusual behavior in the zero-temperature limit, with rigidity transitions that are controlled by residual stresses and therefore exhibit very different scaling and phenomenology compared to particulate systems. Here we investigate the finite-temperature phase of two-dimensional Voronoi and Vertex models, and show that they have highly unusual, sub-Arrhenius scaling of dynamics with temperature. We connect the anomalous glassy dynamics to features of the potential energy landscape associated with zero-temperature inherent states.

Analytical expressions describing ion-induced sputter or etch processes generally relate the sputter yield to the surface atomic binding energy $({U}_{\mathrm{sb}})$ for the target material. While straightforward to measure for the crystalline elemental solids, ${U}_{\mathrm{sb}}$ is more complicated to establish for amorphous and multielement materials due to composition-driven variations and incongruent sublimation. In this regard, we show that for amorphous multielement materials, the ion-driven yield can instead be better understood via a consideration of mechanical rigidity and network topology. We first demonstrate a direct relationship between ${U}_{\mathrm{sb}}$, bulk modulus, and ion sputter yield for the elements, and then subsequently prove our hypothesis for amorphous multielement compounds by demonstrating that the same relationships exist between the reactive ion etch (RIE) rate and nanoindentation Young's modulus for a series of $a\text{\ensuremath{-}}\mathrm{Si}{\mathrm{N}}_{x}$:H and $a\text{\ensuremath{-}}\mathrm{Si}{\mathrm{O}}_{x}{\mathrm{C}}_{y}$:H thin films. The impact of network topology is further revealed via application of the Phillips--Thorpe theory of topological constraints, which directly relates the Young's modulus to the mean atomic coordination $(\ensuremath{\langle}r\ensuremath{\rangle})$ for an amorphous solid. The combined analysis allows the trends and plateaus in the RIE rate to be ultimately reinterpreted in terms of the atomic structure of the target material through a consideration of $\ensuremath{\langle}r\ensuremath{\rangle}$. These findings establish the important underlying role of mechanical rigidity and network topology in ion--solid interactions and provide additional considerations for the design and optimization of radiation-hard materials in nuclear and outer space environments.

The initial dissolution rate of a series of multicomponent glasses is studied in order to discuss the influence of increasing magnesium content in the glass on this alteration regime and to highlight differences in behavior between calcium- and magnesium-bearing glasses. The application of MD-based topological constraint theory (TCT) is confronted to glass transition temperature (Tg) and initial dissolution rate (r0) on a glass series containing the main oxides of a French nuclear glass (AVM). In addition, a comparison between a reference magnesium-containing nuclear waste glass, AVMV4 and a proposed derived simplified composition N19M8 is performed regarding r0 values. Results indicate a similar behavior in this alteration regime for the two glasses, suggesting that this simple glass might be a good analogue to the complex one. Substituting calcium for magnesium decreases the initial dissolution rate by a factor two in the series, while an overall increase of magnesium leads to an increased dissolution rate. Analyses performed with TCT suggests that magnesium environment is better defined than calcium or sodium and may behave as an intermediate species. Finally, a correlation between the number of constraints per atom and Tg is established, while the model failed to link structural features to r0.

Template compositional behaviour of relaxation-time divergence in Ge xSe 1− x