Generic Rigidity Percolation: The Pebble Game

Jacobs, Donald J.; Thorpe, M. F.

doi:10.1103/physrevlett.75.4051

Cited by 458 publications

(570 citation statements)

References 23 publications

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“…Further details about the underlying rigidity theory and the network parameterization have been described elsewhere. 41,57,58 Given a network representation of the RNA structure, the pebble game, 59 a fast combinatorial algorithm, is applied to exactly enumerate the number and spatial distribution of bond rotational degrees of freedom in the network. Based on the accessibility of rotational degrees of freedom, each bond is identified as part of either a rigid cluster or a flexible link in between.…”

Section: Methodsmentioning

confidence: 99%

HIV-1 TAR RNA Spontaneously Undergoes Relevant Apo-to-Holo Conformational Transitions in Molecular Dynamics and Constrained Geometrical Simulations

Fulle

Christ

Kestner

et al. 2010

J. Chem. Inf. Model.

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We report all-atom molecular dynamics and replica exchange molecular dynamics simulations on the unbound human immunodeficiency virus type-1 (HIV-1) transactivation responsive region (TAR) RNA structure and three TAR RNA structures in bound conformations of, in total, ∼250 ns length. We compare the extent of observed conformational sampling with that of the conceptually simpler and computationally much cheaper constrained geometrical simulation approach framework rigidity optimized dynamic algorithm (FRODA). Atomic fluctuations obtained by replica-exchange molecular dynamics (REMD) simulations agree quantitatively with those obtained by molecular dynamics (MD) and FRODA simulations for the unbound TAR structure. Regarding the stereochemical quality of the generated conformations, backbone torsion angles and puckering modes of the sugar-phosphate backbone were reproduced equally well by MD and REMD simulations, but further improvement is needed in the case of FRODA simulations. Essential dynamics analysis reveals that all three simulation approaches show a tendency to sample bound conformations when starting from the unbound TAR structure, with MD and REMD simulations being superior with respect to FRODA. These results are consistent with the experimental view that bound TAR RNA conformations are transiently sampled in the free ensemble, following a conformation selection model. The simulation-generated TAR RNA conformations have been successfully used as receptor structures for docking. This finding has important implications for RNA-ligand docking in that docking into an ensemble of simulation-generated RNA structures is shown to be a valuable means to cope with large apo-to-holo conformational transitions of the receptor structure.

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Section: Methodsmentioning

confidence: 99%

HIV-1 TAR RNA Spontaneously Undergoes Relevant Apo-to-Holo Conformational Transitions in Molecular Dynamics and Constrained Geometrical Simulations

Fulle

Christ

Kestner

et al. 2010

J. Chem. Inf. Model.

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“…placed on the sites (bonds) in a manner analogous to the "pebble game" 4 . This can be found by defining a bulk unit cell with basis vectorsd a that accommodates the surface with no leftover sites or bonds (see Fig.…”

Section: Index Theoremmentioning

confidence: 99%

“…Our starting point is equation (4), which describes the zero-mode count in a region S of a larger system. Using the fact that {H,…”

Section: Supplementary Online Materials Proof Of Index Theoremmentioning

confidence: 99%

Topological boundary modes in isostatic lattices

2013

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Frames, or lattices consisting of mass points connected by rigid bonds or central-force springs, are important model constructs that have applications in such diverse fields as structural engineering, architecture, and materials science. The difference between the number of bonds and the number of degrees of freedom of these lattices determines the number of their zero-frequency "floppy modes". When these are balanced, the system is on the verge of mechanical instability and is termed isostatic. It has recently been shown that certain extended isostatic lattices exhibit floppy modes localized at their boundary. These boundary modes are insensitive to local perturbations, and appear to have a topological origin, reminiscent of the protected electronic boundary modes that occur in the quantum Hall effect and in topological insulators. In this paper we establish the connection between the topological mechanical modes and the topological band theory of electronic systems, and we predict the existence of new topological bulk mechanical phases with distinct boundary modes. We introduce model systems in one and two dimensions that exemplify this phenomenon.Isostatic lattices provide a useful reference point for understanding the properties of a wide range of systems on the verge of mechanical instability, including network glasses 1,2 , randomly diluted lattices near the rigidity percolation threshold 3,4 , randomly packed particles near their jamming threshold 5-10 , and biopolymer networks [11][12][13][14] . There are many periodic lattices, including the square and kagome lattices in d = 2 dimensions and the cubic and pyrochlore lattices in d = 3, that are locally isostatic with coordination number z = 2d for every site under periodic boundary conditions. These lattices, which are the subject of this paper, have a surprisingly rich range of elastic responses and phonon structures [15][16][17][18][19] that exhibit different behaviors as bending forces or additional bonds are added.The analysis of such systems dates to an 1864 paper by James Clerk Maxwell 20 that argued that a lattice with N s mass points and N b bonds has N 0 = dN s − N b zero modes. Maxwell's count is incomplete, though, because N 0 can exceed dN s − N b if there are N ss states of self-stress, where springs can be under tension or compression with no net forces on the masses. This occurs, for example, when masses are connected by straight lines of bonds under periodic boundary conditions. A more general Maxwell relation 21 ,is valid for infinitesimal distortions. In a locally isostatic system with periodic boundary conditions, N 0 = N ss . The square and kagome lattices have one state of self-stress per straight line of bonds and associated zero modes along lines in momentum space. Cutting a section of N sites from these lattices removes states of selfstress and O( √ N ) bonds and necessarily leads to O( √ N ) zero modes, which are essentially identical to the bulk zero modes. Recently Sun et al. 22 studied a twisted kagome lattice in which states ...

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“…It is interesting to note that at first sight the fact that G/B goes to zero at the jamming point as ∆Z, may seem to be the anomalous behavior; however, from the point of view of rigidity percolation [30], B/k behaves anomalously for packings, as in rigidity percolation both G/k and B/k vanish as ∆Z, upon approaching the isostatic point so that G/B remains constant at the percolation threshold. The difference between the jamming transition and generic rigidity percolation is that in the former case configurations must be such that all the contact forces must be repulsive.…”

Section: Behavior or Elastic Constants Near Jammingmentioning

confidence: 99%

The jamming scenario—an introduction and outlook

Liu

Nagel

Saarloos

et al. 2011

Dynamical Heterogeneities in Glasses, Colloids, and Granular Media

119

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Abstract. The jamming scenario of disordered media, formulated about 10 years ago, has in recent years been advanced by analyzing model systems of granular media. This has led to various new concepts that are increasingly being explored in in a variety of systems. This chapter contains an introductory review of these recent developments and provides an outlook on their applicability to different physical systems and on future directions. The first part of the paper is devoted to an overview of the findings for model systems of frictionless spheres, focussing on the excess of low-frequency modes as the jamming point is approached. Particular attention is paid to a discussion of the cross-over frequency and length scales that govern this approach. We then discuss the effects of particle asphericity and static friction, the applicability to bubble models for wet foams in which the friction is dynamic, the dynamical arrest in colloids, and the implications for molecular glasses.

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Generic Rigidity Percolation: The Pebble Game

Cited by 458 publications

References 23 publications

HIV-1 TAR RNA Spontaneously Undergoes Relevant Apo-to-Holo Conformational Transitions in Molecular Dynamics and Constrained Geometrical Simulations

HIV-1 TAR RNA Spontaneously Undergoes Relevant Apo-to-Holo Conformational Transitions in Molecular Dynamics and Constrained Geometrical Simulations

Topological boundary modes in isostatic lattices

The jamming scenario—an introduction and outlook

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