Topology of molecular deformations induces triphasic catch bonding in selectin-ligand bonds.

Among the long-standing efforts to elucidate the physical mechanisms of protein-ligand catch bonding, particular attention has been directed at the family of selectin proteins. Selectins exhibit slip, catch-slip, and slip-catch-slip bonding, with minor structural modifications causing major changes in selectins' response to force. How can a single structural mechanism allow interconversion between these various behaviors? We present a unifying theory of selectin-ligand catch bonding, using a structurally motivated free energy landscape to show how the topology of force-induced deformations of the molecular system produces the full range of observed behaviors. We find that the pathway of bond rupture deforms in non-trivial ways, such that unbinding dynamics depend sensitively on force. This implies a severe breakdown of Bell's theory-a paradigmatic theory used widely in catch bond modeling-raising questions about the suitability of Bell's theory in modeling other catch bonds. Our approach can be applied broadly to other protein-ligand systems.

Physics of viral dynamics.

Viral capsids are often regarded as inert structural units, but in actuality they display fascinating dynamics during different stages of their life cycle. With the advent of single-particle approaches and high-resolution techniques, it is now possible to scrutinize viral dynamics during and after their assembly and during the subsequent development pathway into infectious viruses. In this Review, the focus is on the dynamical properties of viruses, the different physical virology techniques that are being used to study them, and the physical concepts that have been developed to describe viral dynamics.

Kirigami and the Caspar-Klug construction for viral shells with negative Gauss curvature.

In this work we extend the Caspar-Klug construction to the archaeal viruses, which in recent years have captured the attention of many researchers for their ability to thrive in extreme environments. We assume that the shells of archaeal viruses are composed of hexamers and pentamers-as is true for icosahedral viruses-together with heptamers, necessary to introduce negative Gauss curvature. Following the original work of Caspar and Klug, we first construct models capable of reproducing the shape observed in electron microscopy images of archaeal viruses. Next, using the technique of kirigami, we present a systematic way to formulate archaeal virus templates from regular hexagonal lattices. Finally, we utilize the presented techniques to build finite element models of archaeal virus geometries and investigate their shapes as a function of material properties. In particular, using thin-shell elasticity theory, we describe a buckling transition as a function of a modified Föppl-von Kármán number γ^{★} and we show how changes in γ^{★} may initiate the tail formation in the Acidianus two-tailed archaeal virus.

Ground state instabilities of protein shells are eliminated by buckling.

We propose a hybrid discrete-continuum model to study the ground state of protein shells. The model allows for shape transformation of the shell and buckling transitions as well as the competition between states with different symmetries that characterize discrete particle models with radial pair potentials. Our main results are as follows. For large Föppl-von Kármán (FvK) numbers the shells have stable isometric ground states. As the FvK number is reduced, shells undergo a buckling transition resembling that of thin-shell elasticity theory. When the width of the pair potential is reduced below a critical value, then buckling coincides with the onset of structural instability triggered by over-stretched pair potentials. Chiral shells are found to be more prone to structural instability than achiral shells. It is argued that the well-width appropriate for protein shells lies below the structural instability threshold. This means that the self-assembly of protein shells with a well-defined, stable structure is possible only if the bending energy of the shell is sufficiently low so that the FvK number of the assembled shell is above the buckling threshold.

Rheology of semiflexible bundle networks with transient linkers.

We present a theoretical and computational analysis of the rheology of networks made up of bundles of semiflexible filaments bound by transient cross-linkers. Such systems are ubiquitous in the cytoskeleton and can be formed in vitro using filamentous actin and various cross-linkers. We find that their high-frequency rheology is characterized by a scaling behavior that is quite distinct from that of networks of the well-studied single semiflexible filaments. This regime can be understood theoretically in terms of a length-scale-dependent bending modulus for bundles. Next, we observe new dissipative dynamics associated with the shear-induced disruption of the network at intermediate frequencies. Finally, at low frequencies, we encounter a region of non-Newtonian rheology characterized by power-law scaling. This regime is dominated by bundle dissolution and large-scale rearrangements of the network driven by equilibrium thermal fluctuations.

Assembly of viral capsids, buckling, and the Asaro-Grinfeld-Tiller instability.

Icosahedral viral shells are characterized by intrinsic elastic stress focused on the 12 structurally required pentamers. We show that, according to thin-shell theory, assembling icosahedral viral shells should be subject to the Asaro-Grinfeld-Tiller instability (AGTI). AGTIs are encountered in growing epitaxial films exposed to extrinsic elastic stress. For viral shells, the AGTI relieves intrinsic elastic stresses by generating corrugation along the perimeter of the assembling shell. The buckling transition of Lidmar, Mirny, and Nelson provides an alternative mechanism for stress release, which in principle would allow for avoidance of AGTIs. For system parameters appropriate for viral shells however, the AGTI appears to be unavoidable. The azimuthal stress condensation produced by the AGTI might actually assist assembly by providing a guiding mechanism for the insertion of pentamers during viral assembly.

Rhythmogenic neuronal networks, emergent leaders, and k-cores.

Neuronal network behavior results from a combination of the dynamics of individual neurons and the connectivity of the network that links them together. We study a simplified model, based on the proposal of Feldman and Del Negro (FDN) [Nat. Rev. Neurosci. 7, 232 (2006)], of the preBötzinger Complex, a small neuronal network that participates in the control of the mammalian breathing rhythm through periodic firing bursts. The dynamics of this randomly connected network of identical excitatory neurons differ from those of a uniformly connected one. Specifically, network connectivity determines the identity of emergent leader neurons that trigger the firing bursts. When neuronal desensitization is controlled by the number of input signals to the neurons (as proposed by FDN), the network's collective desensitization--required for successful burst termination--is mediated by k-core clusters of neurons.

DNA condensation by field-induced non-equilibrium noise.

DNA molecules under good solvent conditions condense when subjected to strong AC electrical fields. It is shown that AC electrical fields couple charge fluctuations to molecular shape fluctuations, which produces a non-equilibrium noise source that can dominate over equilibrium thermal noise in the long-wavelength limit. The field-induced excess charge fluctuations amplify the Asakura-Oosawa fluctuation attraction force between neighboring chains, providing a mechanism for field-induced DNA condensation.

Flory theory of the folding of designed RNA molecules.

We construct a Flory theory for the folding of RNA molecules that have a designed, linear ground state.

Assembly of viruses and the pseudo-law of mass action.

The self-assembly of the protein shell ("capsid") of a virus appears to obey the law of mass action (LMA) despite the fact that viral assembly is a nonequilibrium process. In this paper we examine a model for capsid assembly, the "assembly line model," that can be analyzed analytically. We show that, in this model, efficient viral assembly from a supersaturated solution is characterized by a shock front propagating in the assembly configuration space from small to large aggregate sizes. If this shock front can reach the size of assembled capsids, then capsid assembly follows either the LMA or a "pseudo" LMA that describes partitioning of capsid proteins between assembled capsids and a metastable, supersaturated solution of free proteins that decays logarithmically slowly. We show that the applicability of the LMA and the pseudo-LMA is governed by two dimensionless parameters: the dimensionless nucleation rate and the dimensionless line energy of incomplete capsids.

Nucleosome switches.

We present a statistical-mechanical model for the positioning of nucleosomes along genomic DNA molecules as a function of the strength of the binding potential and the chemical potential of the nucleosomes. We show that a significant section of the DNA is composed of two-level nucleosome switching regions where the nucleosome distribution undergoes a localized, first-order transition. The location of the nucleosome switches shows a strong correlation with the location of gene-regulation regions.

Phase diagram of chiral biopolymer Wigner crystals.

We study the statistical mechanics of counterion Wigner crystals associated with hexagonal bundles of chiral biopolymers. We show that, due to spontaneous chiral symmetry breaking induced by frustration, these Wigner crystals would be chiral even if the biopolymers themselves were not chiral. Using a duality transformation of the model onto a "spin-charge" Hamiltonian, we show that melting of the Wigner crystal is due to the unbinding of screw dislocations and that the melting temperature has a singular dependence on the intrinsic chirality of the biopolymers. Finally, we report that, if electrostatic interactions are strongly screened, the counterions can condense in the form of an intermediate achiral Wigner solid phase that melts by the unbinding of fractional topological charges.

Chirality and equilibrium biopolymer bundles.

We use continuum theory to show that chirality is a key thermodynamic control parameter for the aggregation of biopolymers: chirality produces a stable disperse phase of hexagonal bundles under moderately poor solvent conditions, as has been observed in in vitro studies of F actin [O. Pelletier et al., Phys. Rev. Lett. 91, 148102 (2003)]. The large characteristic radius of these chiral bundles is not determined by a mysterious long-range molecular interaction but by in-plane shear elastic stresses generated by the interplay between a chiral torque and an unusual, but universal, nonlinear gauge term in the strain tensor of ordered chains that is imposed by rotational invariance.

Microtubule protofilament number is modulated in a stepwise fashion by the charge density of an enveloping layer.

Microtubules are able to adjust their protofilament (PF) number and, as a consequence, their dynamics and function, to the assembly conditions and presence of cofactors. However, the principle behind such variations is poorly understood. Using synchrotron x-ray scattering and transmission electron microscopy, we studied how charged membranes, which under certain conditions can envelop pre-assembled MTs, regulate the PF number of those MTs. We show that the mean PF number, , is modulated primarily by the charge density of the membranes. decreases in a stepwise fashion with increasing membrane charge density. does not depend on the membrane-protein stoichiometry or the solution ionic strength. We studied the effect of taxol and found that increases logarithmically with taxol/tubulin stoichiometry. We present a theoretical model, which by balancing the electrostatic and elastic interactions in the system accounts for the trends in our findings and reveals an effective MT bending stiffness of order 10-100 k(B)T/nm, associated with the observed changes in PF number.

Failure of viral shells.

We report a combined theoretical and experimental study of the structural failure of viral shells under mechanical stress. We find that discontinuities in the force-indentation curve associated with failure should appear when the so-called Föppl-von Kármán (FvK) number exceeds a critical value. A nanoindentation study of a viral shell subject to a soft-mode instability, where the stiffness of the shell decreases with increasing pH, confirms the predicted onset of failure as a function of the FvK number.

RNA condensation and the wetting transition.

We present a continuum theory for the condensation of large, soluble, single-stranded RNA molecules on attractive substrates. In the mean-field approximation, the theory reduces to the Cahn-de Gennes description of wetting fluids and offers a natural explanation for the development of a sharply defined density profile following a prewetting surface phase transition. This mapping onto the wetting problem can break down because of a capillary instability where the adsorbed film decomposes into a collection of segregated, nonoverlapping molecules.

Frustrated polyelectrolyte bundles and T= 0 Josephson-junction arrays.

We establish a one-to-one mapping between a model for hexagonal polyelectrolyte bundles and a model for two-dimensional, frustrated Josephson-junction arrays. We find that the T = 0 insulator-to-superconductor transition of the quantum system corresponds to a continuous liquid-to-solid transition of the condensed charge in the finite-temperature classical system. We find that the role of the vector potential in the quantum system is played by elastic strain in the classical system. Exploiting this correspondence we show that the transition is accompanied by a spontaneous breaking of a discrete symmetry associated with the chiral patterning of the array and that at the transition the polyelectrolyte bundle adopts a universal response to shear.

Elasticity theory and shape transitions of viral shells.

Recently, continuum elasticity theory has been applied to explain the shape transition of icosahedral viral capsids--single-protein-thick crystalline shells--from spherical to "buckled" or faceted as their radius increases through a critical value determined by the competition between stretching and bending energies of a closed two-dimensional (2D) elastic network. In the present work we generalize this approach to capsids with nonicosahedral symmetries, e.g., spherocylindrical and conical shells. One key additional physical ingredient is the role played by nonzero spontaneous curvature. Another is associated with the special way in which the energy of the 12 topologically required fivefold sites depends on the "background" local curvature of the shell in which they are embedded. Systematic evaluation of these contributions leads to a shape "phase" diagram in which transitions are observed from icosahedral to spherocylindrical capsids as a function of the ratio of stretching to bending energies and of the spontaneous curvature of the 2D protein network. We find that the transition from icosahedral to spherocylindrical symmetry is continuous or weakly first order near the onset of buckling, leading to extensive shape degeneracy. These results are discussed in the context of experimentally observed variations in the shapes of a variety of viral capsids.

Hierarchical self-assembly of actin bundle networks: gels with surface protein skin layers.

The networklike structure of actin bundles formed with the cross-linking protein alpha-actinin has been investigated via x-ray scattering and confocal fluorescence microscopy over a wide range of alpha-actinin/F-actin ratios. We describe the hierarchical structure of bundle gels formed at high ratios. Isotropic actin bundle gels form via cluster-cluster aggregation in the diffusion-limited aggregation regime at high alpha-actinin/actin ratios. This process is clearly observed by confocal fluorescence microscopy. Polylysine is investigated as an alternative bundling agent in the high-ratio regime and the effects of F-actin length are also discussed. One particularly fascinating aspect of this system is the presence of a structured skin layer at the gel/water interface. Confocal microscopy has elucidated the full three-dimensional structure of this layer and revealed several interesting morphologies. The protein skin layer is a micron-scale structure composed of a directed network of bundles and exhibits flat, crumpled, and tubelike shapes. We show that crumpling of the skin layer results from stresses due to the underlying gel. These biologically based geometric structures may detach from the gel, demonstrating potential for the generation of biological scaffolds with defined shapes for applications in cell encapsulation and tissue engineering. We demonstrate manipulation of the skin layer, producing hemispherical structures in solution.

Structural polymorphism of the cytoskeleton: a model of linker-assisted filament aggregation.

The phase behavior of charged rods in the presence of interrod linkers is studied theoretically as a model for the equilibrium behavior underlying the organization of actin filaments by linker proteins in the cytoskeleton. The presence of linkers in the solution modifies the effective interrod interaction and can lead to interfilament attraction. Depending on the composition and physical properties of the system, such as linker-binding energies, filaments will orient either perpendicular or parallel to each other, leading to network-like or bundled structures. We show that such a system can have one of three generic phase diagrams, one dominated by bundles, another by networks, and the third containing both bundle and network-like phases. The first two diagrams can be found over a wide range of interaction energies, whereas the third diagram occurs only for a narrow range. These results provide theoretical understanding of the classification of linker proteins as bundling proteins or crosslinking proteins. In addition, they suggest possible mechanisms by which the cell may control cytoskeletal morphology.