Generic rigidity percolation in two dimensions

Penrose tilings as jammed solids

Penrose tilings form lattices, exhibiting fivefold symmetry and isotropic elasticity, with inhomogeneous coordination much like that of the force networks in jammed systems. Under periodic boundary conditions, their average coordination is exactly four. We study the elastic and vibrational properties of rational approximants to these lattices as a function of unit-cell size N(S) and find that they have of order sqrt[N(S)] zero modes and states of self-stress and yet all their elastic moduli vanish. In their generic form, obtained by randomizing site positions, their elastic and vibrational properties are similar to those of particulate systems at jamming with a nonzero bulk modulus, vanishing shear modulus, and a flat density of states.

Stiffest elastic networks

The rigidity of a network of elastic beams is closely related to its microstructure. We show both numerically and theoretically that there is a class of isotropic networks, which are stiffer than any other isotropic network of same density. The elastic moduli of these stiffest elastic networks are explicitly given. They constitute upper-bounds, which compete or improve the well-known Hashin–Shtrikman bounds. We provide a convenient set of criteria (necessary and sufficient conditions) to identify these networks and show that their displacement field under uniform loading conditions is affine down to the microscopic scale. Finally, examples of such networks with periodic arrangement are presented, in both two and three dimensions. In particular, we present an optimal and isotropic three-dimensional structure which, to our knowledge, is the first one to be presented as such.

Two exactly soluble models of rigidity percolation

We summarize results for two exactly soluble classes of bond-diluted models for rigidity percolation, which can serve as a benchmark for numerical and approximate methods. For bond dilution problems involving rigidity, the number of floppy modes F plays the role of a free energy. Both models involve pathological lattices with two-dimensional vector displacements. The first model involves hierarchical lattices where renormalization group calculations can be used to give exact solutions. Algebraic scaling transformations produce a transition of the second order, with an unstable critical point and associated scaling laws at a mean coordination <r>=4.41, which is above the 'mean field' value <r>=4 predicted by Maxwell constraint counting. The order parameter exponent associated with the spanning rigid cluster geometry is β=0.0775 and that associated with the divergence of the correlation length and the anomalous lattice dimension d is dν=3.533. The second model involves Bethe lattices where the rigidity transition is massively first order by a mean coordination <r>=3.94 slightly below that predicted by Maxwell constraint counting. We show how a Maxwell equal area construction can be used to locate the first-order transition and how this result agrees with simulation results on larger random-bond lattices using the pebble game algorithm.

Jamming graphs: a local approach to global mechanical rigidity

We revisit the concept of minimal rigidity as applied to frictionless, repulsive soft sphere packings in two dimensions with the introduction of the jamming graph. Minimal rigidity is a purely combinatorial property encoded via Laman's theorem in two dimensions. It constrains the global, average coordination number of the graph, for example. However, minimal rigidity does not address the geometry of local mechanical stability. The jamming graph contains both properties of global mechanical stability at the onset of jamming and local mechanical stability. We demonstrate how jamming graphs can be constructed using local moves via the Henneberg construction such that these graphs fall under the jurisdiction of correlated percolation. We then probe how jamming graphs destabilize, or become unjammed, by deleting a bond and computing the resulting rigid cluster distribution. We also study how the system restabilizes with the addition of new contacts and how a jamming graph with extra (redundant) contacts destabilizes. The latter endeavor allows us to probe a disk packing in the rigid phase and uncover a potentially new diverging length scale associated with the random deletion of contacts as compared to the study of cut-out (or frozen-in) subsystems.

Bond percolation in higher dimensions

We collect results for bond percolation on various lattices from two to fourteen dimensions that, in the limit of large dimension d or number of neighbors z, smoothly approach a randomly diluted Erdős-Rényi graph. We include results on bond-diluted hypersphere packs in up to nine dimensions, which show the mean coordination, excess kurtosis, and skewness evolving smoothly with dimension towards the Erdős-Rényi limit.

Effective-medium theory of a filamentous triangular lattice

We present an effective-medium theory that includes bending as well as stretching forces, and we use it to calculate the mechanical response of a diluted filamentous triangular lattice. In this lattice, bonds are central-force springs, and there are bending forces between neighboring bonds on the same filament. We investigate the diluted lattice in which each bond is present with a probability p. We find a rigidity threshold p(b) which has the same value for all positive bending rigidity and a crossover characterizing bending, stretching, and bend-stretch coupled elastic regimes controlled by the central-force rigidity percolation point at p(CF)=/~2/3 of the lattice when fiber bending rigidity vanishes.

Dynamic critical response in damped random spring networks

The isostatic state plays a central role in organizing the response of many amorphous materials. We construct a diverging length scale in nearly isostatic spring networks that is defined both above and below isostaticity and at finite frequencies and relate the length scale to viscoelastic response. Numerical measurements verify that proximity to isostaticity controls the viscosity, shear modulus, and creep of random networks.

Changes in Lysozyme Flexibility upon Mutation Are Frequent, Large and Long-Ranged

We investigate changes in human c-type lysozyme flexibility upon mutation via a Distance Constraint Model, which gives a statistical mechanical treatment of network rigidity. Specifically, two dynamical metrics are tracked. Changes in flexibility index quantify differences within backbone flexibility, whereas changes in the cooperativity correlation quantify differences within pairwise mechanical couplings. Regardless of metric, the same general conclusions are drawn. That is, small structural perturbations introduced by single point mutations have a frequent and pronounced affect on lysozyme flexibility that can extend over long distances. Specifically, an appreciable change occurs in backbone flexibility for 48% of the residues, and a change in cooperativity occurs in 42% of residue pairs. The average distance from mutation to a site with a change in flexibility is 17–20 Å. Interestingly, the frequency and scale of the changes within single point mutant structures are generally larger than those observed in the hen egg white lysozyme (HEWL) ortholog, which shares 61% sequence identity with human lysozyme. For example, point mutations often lead to substantial flexibility increases within the β-subdomain, which is consistent with experimental results indicating that it is the nucleation site for amyloid formation. However, β-subdomain flexibility within the human and HEWL orthologs is more similar despite the lowered sequence identity. These results suggest compensating mutations in HEWL reestablish desired properties.

The functional importance of protein dynamics is universally accepted, making the study of dynamical similarities and differences among proteins of the same function an intriguing problem. While some metrics are likely to be conserved across family, differences are also very common. In previous works we have used a Distance Constraint Model to quantify flexibility differences across sets of orthologous proteins, which reproduce this diversity. In the same manner, this work investigates changes occurring upon individual point mutations. Somewhat surprisingly, the small structural perturbations caused by mutation lead to changes throughout the protein. These changes can be quite large, actually surpassing the scale for differences between ortholog pairs. Moreover, changes in flexibility frequently occur at sites far from the mutation site. These results underscore the sensitivity of protein dynamics in connection with allostery, and help explain why differences across protein families are so common.

Structurally determined directionality identifies the boundary between mobile and immobile domains in a disordered material

The structure and available degrees of freedom of an amorphous configuration can determine the location of dynamic heterogeneities. In the same way, these features can also determine the directionality of the particle motion. In this paper we propose that directionality can be attributed to those particles that only participate in a single unconstrained motion. The consequences of this suggestion in terms of the spatial distribution of particles with high directionality are explored using a random bond model.

Nonlinear effective-medium theory of disordered spring networks

Disordered soft materials, such as fibrous networks in biological contexts, exhibit a nonlinear elastic response. We study such nonlinear behavior with a minimal model for networks on lattice geometries with simple Hookian elements with disordered spring constant. By developing a mean-field approach to calculate the differential elastic bulk modulus for the macroscopic network response of such networks under large isotropic deformations, we provide insight into the origins of the strain stiffening and softening behavior of these systems. We find that the nonlinear mechanics depends only weakly on the lattice geometry and is governed by the average network connectivity. In particular, the nonlinear response is controlled by the isostatic connectivity, which depends strongly on the applied strain. Our predictions for the strain dependence of the isostatic point as well as the strain-dependent differential bulk modulus agree well with numerical results in both two and three dimensions. In addition, by using a mapping between the disordered network and a regular network with random forces, we calculate the nonaffine fluctuations of the deformation field and compare them to the numerical results. Finally, we discuss the limitations and implications of the developed theory.

Laws of granular solids: geometry and topology

In a granular solid, mechanical equilibrium requires a delicate balance of forces at the disordered grain scale. To understand how macroscopic rigidity can emerge in this amorphous solid, it is crucial that we understand how Newton's laws pass from the disordered grain scale to the laboratory scale. In this work, we introduce an exact discrete calculus, in which Newton's laws appear as differential relations at the scale of a single grain. Using this calculus, we introduce gauge variables that describe identically force- and torque-balanced configurations. In a first, intrinsic formulation, we use the topology of the contact network, but not its geometry. In a second, extrinsic formulation, we introduce geometry with the Delaunay triangulation. These formulations show, with exact methods, how topology and geometry in a disordered medium are related by constraints. In particular, we derive Airy's expression for a divergence-free, symmetric stress tensor in two and three dimensions.

Internal stresses, normal modes, and nonaffinity in three-dimensional biopolymer networks

We numerically investigate deformations and modes of networks of semiflexible biopolymers as a function of crosslink coordination number z and strength of bending and stretching energies. In equilibrium filaments are under internal stress, and the networks exhibit shear rigidity below the Maxwell isostatic point. In contrast to two-dimensional networks, ours exhibit nonaffine bending-dominated response in all rigid states, including those near the maximum of z=4 when bending energies are less than stretching ones.

Length scales of dynamic heterogeneities in a network of fluctuating mechanical constraints

We describe a random bond network with fluctuating mechanical constraints for which structural and dynamical length scales are explicitly defined and can be compared with the observable properties of the system. The constraints give well-defined individual structures, describing rigid clusters of particles, and there are distinct structural changes between different configurations. In addition, unambiguous unconstrained motions, including extended collective motions, can be defined within each configuration of constraints. By considering fluctuations of bonds in the network, we can analyze the relaxation of a simple model amorphous material subject to randomly arranged mechanical constraints. The lengths of the collective motions are obtained directly and compared with both the relaxation kinetics and the susceptibility χ(4). Above percolation, the latter quantity is shown to decrease with increasing degree of constraint, a behavior linked, in this paper, to the presence of overconstraint in the network at bond densities beyond the percolation point. Our model system, despite a propensity for highly collective motions, shows that an increasing timescale for relaxation can be accompanied by a decreasing χ(4).

Coherent potential approximation of random nearly isostatic kagome lattice

The kagome lattice has coordination number 4, and it is mechanically isostatic when nearest-neighbor sites are connected by central-force springs. A lattice of N sites has O(√N) zero-frequency floppy modes that convert to finite-frequency anomalous modes when next-nearest-neighbor (NNN) springs are added. We use the coherent potential approximation to study the mode structure and mechanical properties of the kagome lattice in which NNN springs with spring constant κ are added with probability P=Δz/4, where Δz=z-4 and z is the average coordination number. The effective medium static NNN spring constant κ(m) scales as P(2) for P≪κ and as P for P≫κ, yielding a frequency scale ω*~Δz and a length scale l*~(Δz)(-1). To a very good approximation at small nonzero frequency, κ(m)(P,ω)/κ(m)(P,0) is a scaling function of ω/ω*. The Ioffe-Regel limit beyond which plane-wave states become ill-defined is reached at a frequency of order ω*.

Hierarchical models of rigidity percolation

We introduce models of generic rigidity percolation in two dimensions on hierarchical networks and solve them exactly by means of a renormalization transformation. We then study how the possibility for the network to self organize in order to avoid stressed bonds may change the phase diagram. In contrast to what happens on random graphs and in some recent numerical studies at zero temperature, we do not find a true intermediate phase separating the usual rigid and floppy ones.

Unconstrained motions, dynamic heterogeneities, and relaxation in disordered solids

A disordered network of bonds with a fixed configuration can relax via a variety of unconstrained motions. These motions can be directly inferred from the topological arrangement of constraints without any geometrical information. We use the pebble game algorithm of Jacobs and Thorpe [D. J. Jacobs and M. F. Thorpe, Phys. Rev. Lett. 75, 4051 (1995)] to decompose the system into separate rigid clusters and identify the remaining degrees of freedom. Unconstrained motions can then be resolved in the form of translations and rotations of isolated groups of bonds and the internal motion within bond groups. We show that each motion can be assigned a characteristic thermal velocity and hence a relaxation time scale. We use this information to construct a relaxation function and also examine the spatial distribution of relaxation time scales. We investigate the sensitivity of the relaxation time scales and their spatial distribution when making individual bond changes in the system, and we consider the dependence of these time scales on the underlying structure.

Rigidity percolation and the spatial heterogeneity of soft modes in disordered materials

We show how the spatial character of unconstrained motion in a network of bonds can be directly inferred from the topological arrangement of constraints. Relaxation time scales of these soft modes are determined, and from this information we generate spatial maps of the heterogeneous distribution of relaxation times in the disordered network. We show that the nature of the dynamic heterogeneity and its sensitivity to changes in bond configuration depends dramatically on the proximity of the system to the rigidity percolation point.

On the collapse of locally isostatic networks

We examine the flexibility of periodic planar networks built from rigid corner-connected equilateral triangles. Such systems are locally isostatic, since for each triangle the total number of degrees of freedom equals the total number of constraints. These nets are two-dimensional analogues of zeolite frameworks, which are periodic assemblies of corner-sharing tetrahedra. If the corner connections are permitted to rotate, as if pin-jointed, there is always at least one collapse mechanism in two dimensions (and at least three mechanisms in three dimensions). We present a number of examples of such collapse modes for different topologies of triangular net. We show that the number of collapse mechanisms grows with the size of unit cell. The collapsible mechanisms that preserve higher symmetry of the network tend to exhibit the widest range of densities without sterical overlap.

Fitting low-resolution cryo-EM maps of proteins using constrained geometric simulations

Recent experimental advances in producing density maps from cryo-electron microscopy (cryo-EM) have challenged theorists to develop improved techniques to provide structural models that are consistent with the data and that preserve all the local stereochemistry associated with the biomolecule. We develop a new technique that maintains the local geometry and chemistry at each stage of the fitting procedure. A geometric simulation is used to drive the structure from some appropriate starting point (a nearby experimental structure or a modeled structure) toward the experimental density, via a set of small incremental motions. Structural motifs such as alpha-helices can be held rigid during the fitting procedure as the starting structure is brought into alignment with the experimental density. After validating this procedure on simulated data for adenylate kinase and lactoferrin, we show how cryo-EM data for two different GroEL structures can be fit using a starting x-ray crystal structure. We show that by incorporating the correct local stereochemistry in the modeling, structures can be obtained with effective resolution that is significantly higher than might be expected from the nominal cryo-EM resolution.

Algorithms for three-dimensional rigidity analysis and a first-order percolation transition

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.

Docking of photosystem I subunit C using a constrained geometric simulation

The elucidation of assembly pathways of multi-subunit protein complexes is a problem of great interest in structural biology and biomolecular modeling. In this study, we use a new computer algorithm for the simulation of large-scale motion in proteins to dock the subunit PsaC onto Photosystem I. We find that a complicated docking pathway involving multiple conformational changes can be quickly simulated by actively targeting only a few residues at a time to their target positions. Simulations for two possible docking scenarios are explored, and experimental approaches to distinguish between them are discussed.

Headgroup Percolation and Collapse of Condensed Langmuir Monolayers

We present a study of Langmuir isotherms and 2D bulk moduli of binary lipid mixtures, where changes in monolayer collapse pressure (Pic) are followed while varying the relative amounts of the two components. For monolayers containing dipalmitoylphosphocholine (DPPC) with either hexadecanol (HD) or palmitic acid (PA), a distinctly non-monotonic change in Pic is observed with varying composition. At low mole fractions, there is a slight decrease in Pic as films get richer in DPPC, while a sharp increase to pure DPPC-like values is observed when the mole fraction exceeds approximately 0.7. The sudden transition in collapse pressure is explained using the principles of rigidity percolation, and important ramifications of this phenomenon for biological surfactant are discussed.

Multiscale modeling of macromolecular conformational changes combining concepts from rigidity and elastic network theory

The development of a two-step approach for multiscale modeling of macromolecular conformational changes is based on recent developments in rigidity and elastic network theory. In the first step, static properties of the macromolecule are determined by decomposing the molecule into rigid clusters by using the graph-theoretical approach FIRST and an all-atom representation of the protein. In this way, rigid clusters are not limited to consist of residues adjacent in sequence or secondary structure elements as in previous studies. Furthermore, flexible links between rigid clusters are identified and can be modeled as such subsequently. In the second step, dynamical properties of the molecule are revealed by the rotations-translations of blocks approach (RTB) using an elastic network model representation of the coarse-grained protein. In this step, only rigid body motions are allowed for rigid clusters, whereas links between them are treated as fully flexible. The approach was tested on a data set of 10 proteins that showed conformational changes on ligand binding. For efficiency, coarse-graining the protein results in a remarkable reduction of memory requirements and computational times by factors of 9 and 27 on average and up to 25 and 125, respectively. For accuracy, directions and magnitudes of motions predicted by our approach agree well with experimentally determined ones, despite embracing in extreme cases >50% of the protein into one rigid cluster. In fact, the results of our method are in general comparable with when no or a uniform coarse-graining is applied; and the results are superior if the movement is dominated by loop or fragment motions. This finding indicates that explicitly distinguishing between flexible and rigid regions is advantageous when using a simplified protein representation in the second step. Finally, motions of atoms in rigid clusters are also well predicted by our approach, which points to the need to consider mobile protein regions in addition to flexible ones when modeling correlated motions.

Constrained geometric simulation of diffusive motion in proteins

We describe a new computational method, FRODA (framework rigidity optimized dynamic algorithm), for exploring the internal mobility of proteins. The rigid regions in the protein are first determined, and then replaced by ghost templates which are used to guide the movements of the atoms in the protein. Using random moves, the available conformational phase space of a 100 residue protein can be well explored in approximately 10-100 min of computer time using a single processor. All of the covalent, hydrophobic and hydrogen bond constraints are maintained, and van der Waals overlaps are avoided, throughout the simulation. We illustrate the results of a FRODA simulation on barnase, and show that good agreement is obtained with nuclear magnetic resonance experiments. We additionally show how FRODA can be used to find a pathway from one conformation to another. This directed dynamics is illustrated with the protein dihydrofolate reductase.

Change in protein flexibility upon complex formation: Analysis of Ras-Raf using molecular dynamics and a molecular framework approach

Changes in flexibility upon protein-protein complex formation of H-Ras and the Ras-binding domain of C-Raf1 have been investigated using the molecular framework approach FIRST (Floppy Inclusion and Rigid Substructure Topology) and molecular dynamics simulations (MD) of in total approximately 35 ns length. In a computational time of about one second, FIRST identifies flexible and rigid regions in a single, static three-dimensional molecular framework, whose vertices represent protein atoms and whose edges represent covalent and non-covalent (hydrogen bond and hydrophobic) constraints and fixed bond angles within the protein. The two methods show a very good agreement with respect to the identification of changes in flexibility in both binding partners on a local scale. This implies that flexibility can be successfully predicted by identifying which bonds limit motion within a molecule and how they are coupled. In particular, as identified by MD, the beta-sheet in Raf shows considerably more pronounced orientational correlations in the bound state compared to the unbound state. Similarly, FIRST assigns the beta-sheet to the largest rigid cluster of the complex. Interestingly, FIRST allows us to identify that interactions across the interface (but not conformational changes upon complex formation) result in the observed rigidification. Since regions of the beta-sheet of Raf that do not interact directly with Ras become rigidified, this also demonstrates the long-range aspect to rigidity percolation. Possible implications of the change of flexibility of the Ras-binding domain of Raf on the activation of Raf upon complex formation are discussed. Finally, the sensitivity of FIRST results with respect to the representation of non-covalent interactions used as constraints is probed.

Rigidity percolation in a field

Rigidity percolation with g degrees of freedom per site is analyzed on randomly diluted Erdös-Renyi graphs, with average connectivity gamma, in the presence of a field h. In the (gamma,h) plane, the rigid and flexible phases are separated by a line of first-order transitions whose location is determined exactly. This line ends at a critical point with classical critical exponents. Analytic expressions are given for the densities n(F) of uncanceled degrees of freedom and gamma(r) of redundant bonds. Upon crossing the coexistence line, gamma(r) and n(F) are continuous, although their first derivatives are discontinuous. We extend, for the case of nonzero field, a recently proposed hypothesis, namely, that the density of uncanceled degrees of freedom is a "free energy" for rigidity percolation. Analytic expressions are obtained for the energy, entropy, and specific heat. Some analogies with a liquid-vapor transition are discussed. Particularizing to zero field, we find that the existence of a (g+1) core is a necessary condition for rigidity percolation with g degrees of freedom. At the transition point gamma(c), Maxwell counting of degrees of freedom is exact on the rigid cluster and on the (g+1) rigid core, i.e., the average coordination of these subgraphs is exactly 2g, although gamma(c), the average coordination of the whole system, is smaller than 2g. gamma(c) is found to converge to 2g for large g, i.e., in this limit Maxwell counting is exact globally as well.

Elasticity of randomly diluted central force networks under tension

We study the rigidity of two-dimensional site-diluted central force triangular networks under tension. We calculate the shear modulus micro directly and fit it with a power law of the form mu approximately (p-p*)(f), where p is the concentration of sites, p* its critical value, and f the critical exponent. We find that the critical behavior of mu is quite sensitive to tension. As the tension is increased there is at first a sharp drop in the values of both p* and f, followed by a slower decrease towards the values of the diluted Gaussian spring network (or random resistor network). We find that the size of the critical region is also sensitive to tension. The tension-free system has a narrower critical regime with the power law failing for p>0.8. In contrast, a small tension is sufficient to extend the power law to near p=1. The physical basis for these behaviors is discussed.

Jamming at zero temperature and zero applied stress: the epitome of disorder

We have studied how two- and three-dimensional systems made up of particles interacting with finite range, repulsive potentials jam (i.e., develop a yield stress in a disordered state) at zero temperature and zero applied stress. At low packing fractions phi, the system is not jammed and each particle can move without impediment from its neighbors. For each configuration, there is a unique jamming threshold phi(c) at which particles can no longer avoid each other, and the bulk and shear moduli simultaneously become nonzero. The distribution of phi(c) values becomes narrower as the system size increases, so that essentially all configurations jam at the same packing fraction in the thermodynamic limit. This packing fraction corresponds to the previously measured value for random close packing. In fact, our results provide a well-defined meaning for "random close packing" in terms of the fraction of all phase space with inherent structures that jam. The jamming threshold, point J, occurring at zero temperature and applied stress and at the random-close-packing density, has properties reminiscent of an ordinary critical point. As point J is approached from higher packing fractions, power-law scaling is found for the divergence of the first peak in the pair correlation function and in the vanishing of the pressure, shear modulus, and excess number of overlapping neighbors. Moreover, near point J, certain quantities no longer self-average, suggesting the existence of a length scale that diverges at J. However, point J also differs from an ordinary critical point: the scaling exponents do not depend on dimension but do depend on the interparticle potential. Finally, as point J is approached from high packing fractions, the density of vibrational states develops a large excess of low-frequency modes. Indeed, at point J, the density of states is a constant all the way down to zero frequency. All of these results suggest that point J is a point of maximal disorder and may control behavior in its vicinity-perhaps even at the glass transition.

Dynamic rigidity transition

An inflated closed loop (or membrane) is used to demonstrate a dynamic rigidity transition that occurs when impact energy is added to the loop in static equilibrium at zero temperature. The only relevant parameter in this transition is the ratio of the energy needed to collapse the loop and the impact energy. When this ratio is below a threshold value close to unity, the loop collapses into a high-entropy floppy state, and it does not return to the rigid state unless the impact energy can escape. The internal oscillations are in the floppy state dominated by 1/f(2) noise. When the ratio is above the threshold, the loop does not collapse, and the internal oscillations resulting from the impact are dominated by the eigenfrequencies of the stretched membrane. In this state, the loop can bounce for a long time. It is still an open question whether bouncing will eventually vanish or whether a stationary bouncing state will be reached. The dynamic transition between the floppy and the rigid state is discontinuous.

Protein unfolding: rigidity lost

We relate the unfolding of a protein to its loss of structural stability or rigidity. Rigidity and flexibility are well defined concepts in mathematics and physics, with a body of theorems and algorithms that have been applied successfully to materials, allowing the constraints in a network to be related to its deformability. Here we simulate the weakening or dilution of the noncovalent bonds during protein unfolding, and identify the emergence of flexible regions as unfolding proceeds. The transition state is determined from the inflection point in the change in the number of independent bond-rotational degrees of freedom (floppy modes) of the protein as its mean atomic coordination decreases. The first derivative of the fraction of floppy modes as a function of mean coordination is similar to the fraction-folded curve for a protein as a function of denaturant concentration or temperature. The second derivative, a specific heat-like quantity, shows a peak around a mean coordination of = 2.41 for the 26 diverse proteins we have studied. As the protein denatures, it loses rigidity at the transition state, proceeds to a state where just the initial folding core remains stable, then becomes entirely denatured or flexible. This universal behavior for proteins of diverse architecture, including monomers and oligomers, is analogous to the rigid to floppy phase transition in network glasses. This approach provides a unifying view of the phase transitions of proteins and glasses, and identifies the mean coordination as the relevant structural variable, or reaction coordinate, along the unfolding pathway.

Random packings of frictionless particles

We conduct numerical simulations of random packings of frictionless particles at T = 0. The packing fraction where the pressure becomes nonzero is the same as the jamming threshold, where the static shear modulus becomes nonzero. The distribution of threshold packing fractions narrows, and its peak approaches random close packing as the system size increases. For packing fractions within the peak, there is no self-averaging, leading to exponential decay of the interparticle force distribution.

Rigidity of random networks of stiff fibers in the low-density limit

Rigidity percolation is analyzed in two-dimensional random networks of stiff fibers. As fibers are randomly added to the system there exists a density threshold q=q(min) above which a rigid stress-bearing percolation cluster appears. This threshold is found to be above the connectivity percolation threshold q=q(c) such that q(min)=(1.1698+/-0.0004)q(c). The transition is found to be continuous, and in the universality class of the two-dimensional central-force rigidity percolation on lattices. At percolation threshold the rigid backbone of the percolating cluster was found to break into rigid clusters, whose number diverges in the limit of infinite system size, when a critical bond is removed. The scaling with system size of the average size of these clusters was found to give a new scaling exponent delta=1.61+/-0.04.

Protein flexibility predictions using graph theory

Techniques from graph theory are applied to analyze the bond networks in proteins and identify the flexible and rigid regions. The bond network consists of distance constraints defined by the covalent and hydrogen bonds and salt bridges in the protein, identified by geometric and energetic criteria. We use an algorithm that counts the degrees of freedom within this constraint network and that identifies all the rigid and flexible substructures in the protein, including overconstrained regions (with more crosslinking bonds than are needed to rigidify the region) and underconstrained or flexible regions, in which dihedral bond rotations can occur. The number of extra constraints or remaining degrees of bond-rotational freedom within a substructure quantifies its relative rigidity/flexibility and provides a flexibility index for each bond in the structure. This novel computational procedure, first used in the analysis of glassy materials, is approximately a million times faster than molecular dynamics simulations and captures the essential conformational flexibility of the protein main and side-chains from analysis of a single, static three-dimensional structure. This approach is demonstrated by comparison with experimental measures of flexibility for three proteins in which hinge and loop motion are essential for biological function: HIV protease, adenylate kinase, and dihydrofolate reductase.

Fractures in heterogeneous two-dimensional systems

A two-dimensional triangular lattice with bond disorder is used as a testing ground for fracture behavior in heterogeneous materials in strain-controlled conditions. Simulations are performed with two interaction potentials (harmonic and Lennard-Jones types) and different breaking thresholds. We study the strain range where the fracture progressively develops from the first to the last breakdown. Scaling properties with the lattice size are investigated: no qualitative difference is found between the two interaction potentials. Clustering properties of the broken bonds are also studied by grouping them into disjoint sets of connected bonds. Finally, the role of kinetic energy is analyzed by comparing overdamped with dissipationless dynamics.

Rigidity transition in two-dimensional random fiber networks

Rigidity percolation is analyzed in two-dimensional random fibrous networks. The model consists of central forces between the adjacent crossing points of the fibers. Two strategies are used to incorporate rigidity: adding extra constraints between second-nearest crossing points with a probability p(sn), and "welding" individual crossing points by adding there four additional constraints with a probability p(weld), and thus fixing the angles between the fibers. These additional constraints will make the model rigid at a critical probability p(sn)=p(sn)(c) and p(weld)=p(weld)(c), respectively. Accurate estimates are given for the transition thresholds and for some of the associated critical exponents. The transition is found in both cases to be in the same universality class as that of the two-dimensional central-force rigidity percolation in diluted lattices.

Rigorous solution for the elasticity of diluted gaussian spring networks

We present a rigorous solution of the elasticity of the diluted Gaussian spring networks (DGSNs) at zero temperature. We show that the deformation of a diluted DGSN is homogeneous provided that the displacements of the particles on the boundary are homogeneous. It follows that at zero temperature the nonvanishing elastic stiffness coefficients are proportional to the hydrostatic pressure in both two and three dimensions. Follows a rigorous proof of the equivalence of the elasticity of the DGSN and the conductance of the random resistor network at zero temperature.

Directed rigidity and bootstrap percolation in 1+1 dimensions

We study directed rigidity percolation (equivalent to directed bootstrap percolation) on three different lattices: square, triangular, and augmented triangular. The first two of these display a first-order transition at p=1, while the augmented triangular lattice shows a continuous transition at a nontrivial p(c). On the augmented triangular lattice we find, by extensive numerical simulation, that the the directed rigidity percolation transition belongs to the same universality class as the directed percolation. The same conclusion is reached by studying its surface critical behavior, i.e., the spreading of rigidity from finite clusters close to a nonrigid wall. Near the discontinuous transition at p=1 on the triangular lattice, we are able to calculate the finite-size behavior of the density of rigid sites analytically. Our results are confirmed by numerical simulation.

Active clusters in disordered systems

We introduce an exact algorithm to calculate the distribution of large low energy clusters (droplets) in disordered manifolds and disordered magnets, and we analyze the extent to which these clusters can be treated as independent two-level systems. We show that interfaces in randomly diluted networks always have broad droplet distributions, while diluted antiferromagnets in a field can have either power law or exponential droplet distributions.

Entropic rigidity of randomly diluted two- and three-dimensional networks

In recent work, we presented evidence that site-diluted triangular central-force networks, at finite temperatures, have a nonzero shear modulus for all concentrations of particles above the geometric percolation concentration p(c). This is in contrast to the zero-temperature case where the (energetic) shear modulus vanishes at a concentration of particles p(r)>p(c). In the present paper we report on analogous simulations of bond-diluted triangular lattices, site-diluted square lattices, and site-diluted simple-cubic lattices. We again find that these systems are rigid for all p>p(c) and that near p(c) the shear modulus mu approximately (p-p(c))(f), where the exponent f approximately 1.3 for two-dimensional lattices and f approximately 2 for the simple-cubic case. These results support the conjecture of de Gennes that the diluted central-force network is in the same universality class as the random resistor network. We present approximate renormalization group calculations that also lead to this conclusion.