Correction: Path Similarity Analysis: A Method for Quantifying Macromolecular Pathways.

[This corrects the article DOI: 10.1371/journal.pcbi.

A broader view on jamming: from spring networks to circle packings.

Jamming occurs when objects like grains are packed tightly together (e.g. grain silos).

Ring correlations in random networks.

We examine the correlations between rings in random network glasses in two dimensions as a function of their separation. Initially, we use the topological separation (measured by the number of intervening rings), but this leads to pseudo-long-range correlations due to a lack of topological charge neutrality in the shells surrounding a central ring. This effect is associated with the noncircular nature of the shells.

Anchored boundary conditions for locally isostatic networks.

Finite pieces of locally isostatic networks have a large number of floppy modes because of missing constraints at the surface. Here we show that by imposing suitable boundary conditions at the surface the network can be rendered effectively isostatic. We refer to these as anchored boundary conditions.

Path Similarity Analysis: A Method for Quantifying Macromolecular Pathways.

Diverse classes of proteins function through large-scale conformational changes and various sophisticated computational algorithms have been proposed to enhance sampling of these macromolecular transition paths. Because such paths are curves in a high-dimensional space, it has been difficult to quantitatively compare multiple paths, a necessary prerequisite to, for instance, assess the quality of different algorithms. We introduce a method named Path Similarity Analysis (PSA) that enables us to quantify the similarity between two arbitrary paths and extract the atomic-scale determinants responsible for their differences.

Partial unfolding and refolding for structure refinement: A unified approach of geometric simulations and molecular dynamics.

The most successful protein structure prediction methods to date have been template-based modeling (TBM) or homology modeling, which predicts protein structure based on experimental structures. These high accuracy predictions sometimes retain structural errors due to incorrect templates or a lack of accurate templates in the case of low sequence similarity, making these structures inadequate in drug-design studies or molecular dynamics simulations. We have developed a new physics based approach to the protein refinement problem by mimicking the mechanism of chaperons that rehabilitate misfolded proteins.

Rigidity loss in disordered systems: three scenarios.

We reveal significant qualitative differences in the rigidity transition of three types of disordered network materials: randomly diluted spring networks, jammed sphere packings, and stress-relieved networks that are diluted using a protocol that avoids the appearance of floppy regions. The marginal state of jammed and stress-relieved networks are globally isostatic, while marginal randomly diluted networks show both overconstrained and underconstrained regions. When a single bond is added to or removed from these isostatic systems, jammed networks become globally overconstrained or floppy, whereas the effect on stress-relieved networks is more local and limited.

Ring statistics of silica bilayers.

The recent synthesis and characterisation of bilayers of vitreous silica has produced valuable new information on ring sizes and distributions. In this paper, we compare the ring statistics of experimental samples with computer generated samples. The average ring size is fixed at six by topology, but the width, skewness and other moments of the distribution of ring edges are characteristics of particular samples.

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.

Fast calculation of the infrared spectra of large biomolecules.

Vibrational spectra of proteins potentially give insight into biologically significant molecular motion and the proportions of different types of secondary structure. Vibrational spectra can be calculated either from normal modes obtained by diagonalizing the mass-weighted Hessian or from the time autocorrelation function derived from molecular dynamics trajectories. The Hessian matrix is calculated from force fields because it is not practical to calculate the Hessian from quantum mechanics for large molecules.

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.

Amorphous graphene: a realization of Zachariasen's glass.

Amorphous graphene is a realization of a two-dimensional Zachariasen glass as first proposed 80 years ago. Planar continuous random networks of this archetypal two-dimensional network are generated by two complementary simulation methods. In the first, a Monte Carlo bond switching algorithm is employed to systematically amorphize a crystalline graphene sheet.

DNA Bending through Large Angles Is Aided by Ionic Screening.

We used adaptive umbrella sampling on a modified version of the roll angle to simulate the bending of DNA dodecamers. Simulations were carried out with the AMBER and CHARMM force fields for 10 sequences in which the central base pair step was varied. On long length scales, the DNA behavior was found to be consistent with the worm-like chain model.

Collective dynamics differentiates functional divergence in protein evolution.

Protein evolution is most commonly studied by analyzing related protein sequences and generating ancestral sequences through Bayesian and Maximum Likelihood methods, and/or by resurrecting ancestral proteins in the lab and performing ligand binding studies to determine function. Structural and dynamic evolution have largely been left out of molecular evolution studies. Here we incorporate both structure and dynamics to elucidate the molecular principles behind the divergence in the evolutionary path of the steroid receptor proteins.

Protein unfolding under force: crack propagation in a network.

The mechanical unfolding of a set of 12 proteins with diverse topologies is investigated using an all-atom constraint-based model. Proteins are represented as polypeptides cross-linked by hydrogen bonds, salt bridges, and hydrophobic contacts, each modeled as a harmonic inequality constraint capable of supporting a finite load before breaking. Stereochemically acceptable unfolding pathways are generated by minimally overloading the network in an iterative fashion, analogous to crack propagation in solids.

Comparison of pathways from the geometric targeting method and targeted molecular dynamics in nitrogen regulatory protein C.

Geometric targeting (GT) is a recently introduced method for rapidly generating all-atom pathways from one protein state to another, based on geometric rather than energetic considerations. To generate pathways, a bias is applied that gradually moves atoms toward a target structure, while a set of geometric constraints between atoms is enforced to keep the structure stereochemically acceptable. In this work, we compare conformational pathways generated from GT to pathways from the much more computationally intensive and commonly used targeted molecular dynamics (TMD) technique, for a complicated conformational change in the signaling protein nitrogen regulatory protein C.

Generating stereochemically acceptable protein pathways.

We describe a new method for rapidly generating stereochemically acceptable pathways in proteins. The method, called geometric targeting, is publicly available at the webserver http://pathways.asu.

Flexibility of ideal zeolite frameworks.

We explore the flexibility windows of the 194 presently-known zeolite frameworks. The flexibility window represents a range of densities within which an ideal zeolite framework is stress-free. Here, we consider the ideal zeolite to be an assembly of rigid corner-sharing perfect tetrahedra.

The long-wavelength limit of the structure factor of amorphous silicon and vitreous silica.

Liquids are in thermal equilibrium and have a non-zero structure factor S(Q --> 0) = [-(2)]/ = rho(0)k(B)Tchi(T) in the long-wavelength limit where rho(0) is the number density, T is the temperature, Q is the scattering vector and chi(T) is the isothermal compressibility. The first part of this result involving the number N (or density) fluctuations is a purely geometrical result and does not involve any assumptions about thermal equilibrium or ergodicity, so is obeyed by all materials. From a large computer model of amorphous silicon, local number fluctuations extrapolate to give S(0) = 0.

Hierarchical plasticity from pair distance fluctuations.

Observations, experiments and simulations often generate large numbers of snapshots of configurations of complex many-body systems. It is important to find methods of extracting useful information from these ensembles of snapshots in order to document the motion as the system evolves in time. Some of the most interesting information is contained in the relative motion of individual constituents, rather than their absolute motion.

Quadrupolar ordering in LaMnO3 revealed from scattering data and geometric modeling.

Many strongly correlated materials display quadrupolar (Jahn-Teller) distortion of the local octahedral structural units. It is common for these distortions to be observed by probes of local structure but absent in the crystallographic average structure. The ordering of these quadrupoles is important in determining the properties of manganites and cuprates, and the nature of the disorder in these structures has been an unsolved problem.

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.

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.

Comment on elastic network models and proteins.

Elastic network models have been used to study the properties of coarse grained models of proteins and larger biomolecular complexes. In this comment, we point out that it is important to build rotational symmetry, as well as translational symmetry, into these models that are designed to describe the rigidity, and the associated low-frequency deformations. This leads to strong restrictions on what form of interactions can be used.

Quantum correction to the pair distribution function.

We report a numerical technique that allows the quantum effects of zero-point motion to be incorporated into Pair Distribution Functions calculated classically for molecules using Monte Carlo or Molecular Dynamics simulations. We establish the basis for this approximation using a diatomic molecule described by a Morse potential. The correction should significantly improve the agreement between modeled and experimental data, and facilitate conclusions about inter- and intra-molecular motion and flexibility.