Lamellar phase of stacked two-dimensional rafts of actin filaments

Molecular scale imaging of F-actin assemblies immobilized on a photopolymer surface

A photo-immobilization based process is presented for direct imaging of hierarchical assemblies of biopolymers using atomic force microscopy (AFM). The technique was used to investigate the phase behavior of F-actin aggregates as a function of concentration of the divalent cation Mg2+. The data provided direct experimental evidence of a coil-on-coil (braided) structure of F-actin bundles formed at high Mg2+ concentrations. At intermediate Mg2+ concentrations, the data showed the first images of the two-dimensional nematic rafts discovered by recent x-ray studies and theoretical treatments.

Lattice-gas model of DNA charge inversion by a positively charged polyelectrolyte

The model of DNA charge inversion by Nguyen and Shklovskii [T. T. Nguyen and B. I. Shklovskii, Phys. Rev. Lett. 89, 018101 (2002)] is extended. A single double-helix strand of DNA is represented by a lattice of negative charges at the positions of the protruding oxygens of the phosphates along the DNA backbone, and the adsorbed polyelectrolyte molecules are represented by charged dimers. A lattice-gas model is used in which dimers adsorbing either parallel or perpendicular to the lattice are treated as separate species, and the model allows for vacancies between adsorbed species. The mean field theory used is formulated as a saddle-point approximation of the exact functional integral representation of the grand canonical partition function, opening the way for the inclusion of the effects of charge fluctuation corrections.

Scaling and universality in the counterion-condensation transition at charged cylinders

Counterions at charged rodlike polymers exhibit a condensation transition at a critical temperature (or, equivalently, at a critical linear charge density for polymers), which dramatically influences various static and dynamic properties of charged polymer solutions. We address the critical and universal aspects of this transition for counterions at a single charged cylinder in two and three spatial dimensions using numerical and analytical methods. By introducing a Monte Carlo sampling method in logarithmic radial scale, we are able to numerically simulate the critical limit of infinite system size (corresponding to the infinite-dilution limit) within tractable equilibration times. The critical exponents are determined for the inverse moments of the counterionic density profile (which play the role of the order parameters and represent the mean inverse localization length of counterions) both within mean-field theory and within Monte Carlo simulations. In three dimensions (3D), we demonstrate that correlation effects (neglected within mean-field theory) lead to an excessive accumulation of counterions near the charged cylinder below the critical temperature (i.e., in the condensation phase), while surprisingly, the critical region exhibits universal critical exponents in accordance with mean-field theory. Also in contrast with the typical trend in bulk critical phenomena, where fluctuations become more enhanced in lower dimensions, we demonstrate, using both numerical and analytical approaches, that mean-field theory becomes exact for the two-dimensional (2D) counterion-cylinder system at all temperatures (Manning parameters), when the number of counterions tends to infinity. For a finite number of particles, however, the 2D problem displays a series of peculiar singular points (with diverging heat capacity), which reflect successive delocalization events of individual counterions from the central cylinder. In both 2D and 3D, the heat capacity shows a universal jump at the critical point and the internal energy develops a pronounced peak. The asymptotic behavior of the energy peak location is used to determine the critical temperature, which is also found to be in agreement with the mean-field prediction.

Structural transition of actin filament in a cell-sized water droplet with a phospholipid membrane

Actin filament, F-actin, is a semiflexible polymer with a negative charge, and is one of the main constituents of cell membranes. To clarify the effect of cross talk between a phospholipid membrane and actin filaments in cells, we conducted microscopic observations on the structural changes in actin filaments in a cell-sized (several tens of micrometers in diameter) water droplet coated with a phospholipid membrane such as phosphatidylserine (PS; negatively charged head group) or phosphatidylethanolamine (PE; neutral head group) as a simple model of a living cell membrane. With PS, actin filaments are distributed uniformly in the water phase without adsorption onto the membrane surface between 2 and 6 mM Mg2+, while between 6 and 12 mM Mg2+, actin filaments are adsorbed onto the inner membrane surface. With PE, the actin filaments are uniformly adsorbed onto the inner membrane surface between 2 and 12 mM Mg2+. With both PS and PE membranes, at Mg2+ concentrations higher than 12 mM, thick bundles are formed in the bulk water droplet accompanied by the dissolution of actin filaments from the membrane surface. The attraction between actin filaments and membrane is attributable to an increase in the translational entropy of counterions accompanied by the adsorption of actin filaments onto the membrane surface. These results suggest that a microscopic water droplet coated with phospholipid can serve as an easy-to-handle model of cell membranes.

Condensation of DNA-actin polyelectrolyte mixtures driven by ions of different valences

Multivalent ions can induce condensation of like-charged polyelectrolytes into compact states, a process that requires different ion valences for different polyelectrolyte species. In this work we examine the condensation behavior in binary anionic polyelectrolyte mixtures consisting of DNA coils and F-actin rods in the presence of monovalent, divalent, and trivalent ions. As expected, monovalent ions do not condense either component and divalent ions selectively condense F-actin rods out of the polyelectrolyte mixture. For trivalent ions, however, we observe a microphase separation between the two polyelectrolytes into coexisting finite-sized F-actin bundles and DNA toroids. Further, by increasing the DNA volume fraction in the mixture, condensed F-actin bundles can be completely destabilized, leading to only DNA condensation within the mixture. We examine a number of possible causes and propose a model based on polyelectrolyte competition for ions.

Morphological transitions during the formation of templated mesoporous materials: theoretical modeling

We put forward a theoretical model for the morphological transitions of templated mesoporous materials. These materials consist of a mixture of surfactant molecules and inorganic compounds which evolve dynamically upon mixing to form different morphologies depending on the composition and conditions at which mixing occurs. Our theoretical analysis is based on the assumption that adsorption of the inorganic compounds onto mesoscopic assemblies of surfactant molecules changes the effective interactions between the surfactant molecules, consequently lowering the spontaneous curvature of the surfactant layer and inducing morphological changes in the system. On the basis of a mean field phase diagram, we are able to follow the trajectories of the system starting with different initial conditions, and predict the final morphology of the product. In a typical scenario, the reduction in the spontaneous curvature leads first to a smooth transition from compact spherical micelles to elongated worm-like micelles. In the second stage, the layer of inorganic material coating the micelles gives rise to attractive inter-micellar interactions that eventually induce a collapse of the system into a closely packed hexagonal array of coated cylinders. Other pathways may lead to different structures including disordered bicontinuous and ordered cubic phases. The model is in good qualitative agreement with experimental observations.

Orientational ordering and dynamics of rodlike polyelectrolytes

The interplay between electrostatic interactions and orientational correlations is studied for a model system of charged rods positioned on a chain, using Monte Carlo simulation techniques. It is shown that the coupling brings about the notion of electrostatic frustration, which in turn results in: (i) a rich variety of orientational orderings such as chiral phases, and (ii) an inherently slow dynamics characterized by stretched-exponential behavior in the relaxation functions of the system.

Direct observation of counterion organization in F-actin polyelectrolyte bundles

Attractions between like-charged polyelectrolytes have been observed in a variety of systems (W.M. Gelbart, R.F. Bruinsma, P.A. Pincus, V.A. Parsegian, Phys. Today 53, September issue, 38 (2000)). Recent biological examples include DNA, filamentous viruses, and F-actin. Theoretical investigations on idealized systems indicate that counterion correlations play a central role, but no experiments that specifically probe such correlations have been performed. Using synchrotron X-ray diffraction, we have directly observed the organization of multivalent ions on cytoskeletal filamentous actin (a well-defined biological polyelectrolyte) and found an unanticipated symmetry-breaking collective counterion mechanism for generating attractions. Surprisingly, the counterions do not form a lattice that simply follows actin's helical symmetry; rather, the counterions organize into "frozen" ripples parallel to the actin filaments and form structures reminiscent of charge density waves. Moreover, these 1D counterion charge density waves form a coupled mode with twist deformations of the oppositely charged actin filaments. This counterion organization is not sensitive to thermal fluctuations in temperature range accessible to protein-based polyelectrolyte systems. Moreover, the counterion density waves are "pinned" to the spatial periodicity of charges on the actin filament even if the global filament charge density is varied, indicating the importance of charge periodicity on the polyelectrolyte substrate.

Dielectric control of counterion-induced single-chain folding transition of DNA

In the presence of condensing agents, single chains of giant double-stranded DNA undergo a first-order phase transition between an elongated coil state and a folded compact state. To connect this like-charged attraction phenomenon to counterion condensation, we performed a series of single-chain experiments on aqueous solutions of DNA, where we varied the extent of counterion condensation by varying the relative dielectric constant epsilon(r) from 80 to 170. Single-chain observations of changes in the conformation of giant DNA were performed by transmission electron microscopy and fluorescence microscopy, with tetravalent spermine (SPM(4+)) as a condensing agent. At a fixed dielectric constant, single DNA chains fold into a compact state upon the addition of spermine, whereas at a constant spermine concentration single DNA chains unfold with an increase in epsilon(r). In both cases, the transition is largely discrete at the level of single chains. We found that the critical concentration of spermine necessary to induce the single-chain folding transition increases exponentially as the dielectric constant increases, corresponding to 87-88% of the DNA charge neutralized at the onset of the transition. We also observed that the toroidal morphology of compact DNA partially unfolds when epsilon(r) is increased.

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.

Spatial distribution of competing ions around DNA in solution

The competition of monovalent and divalent cations for proximity to negatively charged DNA is of biological importance and can provide strong constraints for theoretical treatments of polyelectrolytes. Resonant x-ray scattering experiments have allowed us to monitor the number and distribution of each cation in a mixed ion cloud around DNA. These measurements provide experimental evidence to support a general theoretical prediction: the normalized distribution of each ion around polyelectrolytes remains constant when ions are mixed at different ratios. In addition, the amplitudes of the scattering signals throughout the competition provide a measurement of the surface concentration parameter that predicts the competition behavior of these cations. The data suggest that ion size needs to be taken into account in applying Poisson-Boltzmann treatments to polyelectrolytes such as DNA.

Higher-order assembly of microtubules by counterions: from hexagonal bundles to living necklaces

Cellular factors tightly regulate the architecture of bundles of filamentous cytoskeletal proteins, giving rise to assemblies with distinct morphologies and physical properties, and a similar control of the supramolecular organization of nanotubes and nanorods in synthetic materials is highly desirable. However, it is unknown what principles determine how macromolecular interactions lead to assemblies with defined morphologies. In particular, electrostatic interactions between highly charged polyelectrolytes, which are ubiquitous in biological and synthetic self-assembled structures, are poorly understood. We have used a model system consisting of microtubules (MTs) and multivalent cations to examine how microscopic interactions can give rise to distinct bundle phases in biological polyelectrolytes. The structure of these supramolecular assemblies was elucidated on length scales from subnanometer to micrometer with synchrotron x-ray diffraction, transmission electron microscopy, and differential interference contrast microscopy. Tightly packed hexagonal bundles with controllable diameters were observed for large trivalent, tetravalent, and pentavalent counterions. Unexpectedly, in the presence of small divalent cations, we have discovered a living necklace bundle phase, comprised of 2D dynamic assemblies of MTs with linear, branched, and loop topologies. This new bundle phase is an experimental example of nematic membranes. The morphologically distinct MT assemblies give insight into general features of bundle formation and may be used as templates for miniaturized materials with applications in nanotechnology and biotechnology.

Crawling toward a unified model of cell mobility: spatial and temporal regulation of actin dynamics

Crawling cells of various morphologies displace themselves in their biological environments by a similar overall mechanism of protrusion through actin assembly at the front coordinated with retraction at the rear. Different cell types organize very distinct protruding structures, yet they do so through conserved biochemical mechanisms to regulate actin polymerization dynamics and vary the mechanical properties of these structures. The moving cell must spatially and temporally regulate the biochemical interactions of its protein components to exert control over higher-order dynamic structures created by these proteins and global cellular responses four or more orders of magnitude larger in scale and longer in time than the individual protein-protein interactions that comprise them. To fulfill its biological role, a cell globally responds with high sensitivity to a local perturbation or signal and coordinates its many intracellular actin-based functional structures with the physical environment it experiences to produce directed movement. This review attempts to codify some unifying principles for cell motility that span organizational scales from single protein polymer filaments to whole crawling cells.

Effect of mono- and multivalent salts on angle-dependent attractions between charged rods

Using molecular dynamics simulations we examine the effective interactions between two like-charged rods as a function of angle and separation. In particular, we determine how the competing electrostatic repulsions and multivalent-ion-induced attractions depend upon concentrations of simple and multivalent salts. We find that with increasing multivalent salt, the stable configuration of two rods evolves from isolated rods to aggregated perpendicular rods to aggregated parallel rods; at sufficiently high concentration, additional multivalent salt reduces the attraction. Monovalent salt enhances the attraction near the onset of aggregation and reduces it at a higher concentration of multivalent salt.

Skin layer at the actin-gel surface: quenched protein membranes form flat, crumpled, and tubular morphologies

We report the discovery of a hierarchically structured skin layer formed at the surface of an isotropic gel of filamentous actin bundles at high molar ratios of alpha-actinin, an actin cross-linker, to globular actin. Confocal microscopy has elucidated the full, micron scale 3D structure. The protein skin layer, composed of a directed network of bundles, exhibits flat, crumpled, tubelike and pleated multitubular morphologies, resulting from stresses due to the underlying gel. The skin layer, which readily detaches, constitutes a model anisotropic solid membrane with stress-induced, quenched disorder.

Thermoreversible crosslinking of polyelectrolyte chains

Thermoreversible crosslinking of polyelectrolyte chains via short-range attractions such as hydrogen bonding induced by uncharged or charged particles is studied within the Flory model of ideal association. Electrostatic interactions between the charges at different linking fractions are taken into account by using a generalized random phase approximation approach which includes the network connectivity. We find that at certain concentration of linking agents an infinitely large polymer network is formed. We calculate the structural gelation lines for linkers of different charges and functionalities.

Structure of autocatalytically branched actin solutions

The average branching number and cluster size in branched actin solutions with filament capping are evaluated using analytic theory and simulation methods. The average number of daughter branches per filament in steady state is much less than unity, regardless of the concentration of branching stimulant. Much more highly branched structures are obtained in the initial stages of polymerization.

Attraction of like-charged macroions in the strong-coupling limit

Like-charged macroions attract each other as a result of strong electrostatic correlations in the presence of multivalent counterions or at low temperatures. We investigate the effective electrostatic interaction between i) two like-charged rods and ii) two like-charged spheres using the recently introduced strong-coupling theory, which becomes asymptotically exact in the limit of large coupling parameter (i.e. for large counterion valency, low temperature, or high surface charge density on macroions). In contrast to previous applications of the strong-coupling theory, we deal with curved surfaces and an additional parameter, referred to as Manning parameter, is introduced, which measures the ratio between the radius of curvature of macroions to the Gouy-Chapman length. This parameter, together with the size of the confining box enclosing the two macroions and their neutralizing counterions, controls the counterion-condensation process that directly affects the effective interactions. For sufficiently large Manning parameters (weakly-curved surfaces), we find a strong long-ranged attraction between two macroions that form a closely-packed bound state with small surface-to-surface separation of the order of the counterion diameter in agreement with recent simulations results. For small Manning parameters (highly-curved surfaces), on the other hand, the equilibrium separation increases and the macroions unbind from each other as the confinement volume increases to infinity. This occurs via a continuous universal unbinding transition for two charged rods at a threshold Manning parameter of Epsilon c = 2/3, while the transition is strongly discontinuous for spheres because of a pronounced potential barrier at intermediate distances. Unlike the cylindrical case, the attractive forces between spheres disappear slowly for increasing confinement volume due to the complete de-condensation of counterions. Scaling arguments suggest that for moderate values of coupling parameter, strong-coupling predictions remain valid for sufficiently small surface-to-surface separations.

Structure of actin cross-linked with alpha-actinin: a network of bundles

Three-dimensional laser scanning confocal microscopy has revealed that filamentous actin, when complexed with the cross-linking protein alpha-actinin, will spontaneously assemble on a micron scale into a structure comprised of a relatively rigid, frequently branching, 3D network of bundles with characteristic mesh size of the order of the persistence length of F-actin. In contrast, additional nanoscale ordering is observed, as synchrotron x-ray diffraction has revealed a disordered, distorted square lattice of actin fibers within the individual bundles.