Platonic and Archimedean geometries in multicomponent elastic membranes.

Large crystalline molecular shells, such as some viruses and fullerenes, buckle spontaneously into icosahedra. Meanwhile multicomponent microscopic shells buckle into various polyhedra, as observed in many organelles. Although elastic theory explains one-component icosahedral faceting, the possibility of buckling into other polyhedra has not been explored. We show here that irregular and regular polyhedra, including some Archimedean and Platonic polyhedra, arise spontaneously in elastic shells formed by more than one component. By formulating a generalized elastic model for inhomogeneous shells, we demonstrate that coassembled shells with two elastic components buckle into polyhedra such as dodecahedra, octahedra, tetrahedra, and hosohedra shells via a mechanism that explains many observations, predicts a new family of polyhedral shells, and provides the principles for designing microcontainers with specific shapes and symmetries for numerous applications in materials and life sciences.

Charge renormalization of bilayer elastic properties.

By combining molecular dynamics simulations and analytical arguments, we investigate the elastic properties of charged lipid bilayers. We show that electrostatic interactions between the head groups can lead to solidification of the lipid bilayer that would otherwise be in a liquid state if the charges were absent. All elastic parameters of the bilayer such as the bending rigidity κ and the two-dimensional bulk modulus λ and Young's modulus Y are found to depend on the values of the charges assigned to the lipid head groups. To extract κ and λ, we fit the molecular dynamics data to a standard elastic model for lipid bilayers. Moreover, we analytically obtain the dependence of the Young modulus Y on the relative strengths of electrostatic and van der Waals interactions in the zero temperature limit.

Shape change of nanocontainers via a reversible ionic buckling.

We demonstrate that small charged nanocages can undergo reversible changes of shapes by modifying the ionic conditions including salt concentration, pH, and dielectric permittivity of the medium. Using numerical simulations, we analyze structures with various charge stoichiometric ratios. At zero or low charge densities, the shape of the cage is determined by its elastic properties, and the surface charge pattern is dictated by the globally fixed geometry. As the charge density per molecule increases, the shape is strongly affected by the electrostatic forces. In this regime, the shape of the nanocage is controlled by the charge distribution.

Faceting ionic shells into icosahedra via electrostatics.

Shells of various viruses and other closed packed structures with spherical topology exhibit icosahedral symmetry because the surface of a sphere cannot be tiled without defects, and icosahedral symmetry yields the most symmetric configuration with the minimum number of defects. Icosahedral symmetry is different from icosahedral-shaped structures, which include some large viruses, cationic-anionic vesicles, and fullerenes. We present a faceting mechanism of ionic shells into icosahedral shapes that breaks icosahedral symmetry resulting from different arrangements of the charged components among the facets. These self-organized ionic structures may favor the formation of flat domains on curved surfaces. We show that icosahedral shapes without rotational symmetry can have lower energy than spheres with icosahedral symmetry caused by preferred bending directions in the planar ionic lattice. The ability to create icosahedral shapes without icosahedral symmetry may lead to the design of new functional materials. The electrostatically driven faceting mechanism we present here suggests that we can design faceted polyhedra with diverse symmetries by coassembling oppositely charged molecules of different stoichiometric ratios.

Multicanonical Monte Carlo ensemble growth algorithm.

We present an ensemble Monte Carlo growth method to sample the equilibrium thermodynamic properties of random chains. The method is based on the multicanonical technique of computing the density of states in the energy space. Such a quantity is temperature independent, and therefore microcanonical and canonical thermodynamic quantities, including the free energy, entropy, and thermal averages, can be obtained by reweighting with a Boltzmann factor. The algorithm we present combines two approaches: The first is the Monte Carlo ensemble growth method, where a "population" of samples in the state space is considered, as opposed to traditional sampling by long random walks, or iterative single-chain growth. The second is the flat-histogram Monte Carlo, similar to the popular Wang-Landau sampling, or to multicanonical chain-growth sampling. We discuss the performance and relative simplicity of the proposed algorithm, and we apply it to known test cases.

Buckled membranes in mixed-valence ionic amphiphile vesicles.

Amphiphilic molecules can form closed-shell structures that are determined by competing attractive and repulsive forces. Since supramolecular shape has its roots in intermolecular interactions, the interplay of electrostatic, hydrophobic, and steric forces can generate nonspherical structures. Here we show that anionic and cationic amphiphiles of unequal charge can coassemble into small buckled vesicles and present a physical argument that explains this phenomenon. The strong electrostatic interaction between the +3 and -1 head groups increases the cohesion energy of the amphiphiles and favors the formation of two-dimensional, flat ionic domains on the vesicle surface, resulting in edges and a buckled shape.

Low-radii transitions in co-assembled cationic-anionic cylindrical aggregates.

We investigate the formation of charged patterns on the surface of cylindrical micelles from co-assembled cationic and anionic amphiphiles. The competition between the net incompatibility chi (which arises from the different chemical nature of oppositely charged molecules) and electrostatic interactions (which prevent macroscopic segregation) results in the formation of surface domains. We employ Monte Carlo simulations to study the domains at thermal equilibrium. Our results extend previous work by studying the effect of the Bjerrum length l(B) at different values of the cylinder's radius R and chi and analyze how it affects the transition between helical, ring, and isotropic patterns. A critical surface in the space (l(B), R, chi) separating these three phases is found, and we show how it corresponds to a first-order phase transition. This confirms that the Bjerrum length l(B) is a significant parameter in the control of the helical-ring transition; the ring pattern is strongly associated with short-range forces, whereas the helical pattern develops from dominant long-range electrostatic interactions.

Topology of pseudoknotted homopolymers.

We consider the folding of a self-avoiding homopolymer on a lattice, with saturating hydrogen bond interactions. Our goal is to numerically evaluate the statistical distribution of the topological genus of pseudoknotted configurations. The genus has been recently proposed for classifying pseudoknots (and their topological complexity) in the context of RNA folding. We compare our results on the distribution of the genus of pseudoknots, with the theoretical predictions of an existing combinatorial model for an infinitely flexible and stretchable homopolymer. We thus obtain that steric and geometric constraints considerably limit the topological complexity of pseudoknotted configurations, as it occurs for instance in real RNA molecules. We also analyze the scaling properties at large homopolymer length, and the genus distributions above and below the critical temperature between the swollen phase and the compact-globule phase, both in two and three dimensions.

Classification and predictions of RNA pseudoknots based on topological invariants.

We propose a new topological characterization of ribonucleic acid (RNA) secondary structures with pseudoknots based on two topological invariants. Starting from the classic arc representation of RNA secondary structures, we consider a model that couples both (i) the topological genus of the graph and (ii) the number of crossing arcs of the corresponding primitive graph. We add a term proportional to these topological invariants to the standard free energy of the RNA molecule, thus obtaining a novel free-energy parametrization that takes into account the abundance of topologies of RNA pseudoknots observed in RNA databases.

Molecular crystallization controlled by pH regulates mesoscopic membrane morphology.

Coassembled molecular structures are known to exhibit a large variety of geometries and morphologies. A grand challenge of self-assembly design is to find techniques to control the crystal symmetries and overall morphologies of multicomponent systems. By mixing +3 and -1 ionic amphiphiles, we assemble crystalline ionic bilayers in a large variety of geometries that resemble polyhedral cellular crystalline shells and archaea wall envelopes. We combine TEM with SAXS and WAXS to characterize the coassembled structures from the mesoscopic to nanometer scale. The degree of ionization of the amphiphiles and their intermolecular electrostatic interactions are controlled by varying pH. At low and high pH values, we observe closed, faceted vesicles with two-dimensional hexagonal molecular arrangements, and at intermediate pH, we observe ribbons with rectangular-C packing. Furthermore, as pH increases, we observe interdigitation of the bilayer leaflets. Accurate atomistic molecular dynamics simulations explain the pH-dependent bilayer thickness changes and also reveal bilayers of hexagonally packed tails at low pH, where only a small fraction of anionic headgroups is charged. Coarse-grained simulations show that the mesoscale geometries at low pH are faceted vesicles where liquid-like edges separate flat crystalline domains. Our simulations indicate that the curved-to-polyhedral shape transition can be controlled by tuning the tail density in regions where sharp bends can form the polyhedral edges. In particular, the pH acts to control the overall morphology of the ionic bilayers by changing the local crystalline order of the amphiphile tails.

Coulomb interactions in charged fluids.

The use of Ewald summation schemes for calculating long-range Coulomb interactions, originally applied to ionic crystalline solids, is a very common practice in molecular simulations of charged fluids at present. Such a choice imposes an artificial periodicity which is generally absent in the liquid state. In this paper we propose a simple analytical O(N(2)) method which is based on Gauss's law for computing exactly the Coulomb interaction between charged particles in a simulation box, when it is averaged over all possible orientations of a surrounding infinite lattice. This method mitigates the periodicity typical of crystalline systems and it is suitable for numerical studies of ionic liquids, charged molecular fluids, and colloidal systems with Monte Carlo and molecular dynamics simulations.

Electrostatics and optimal arrangement of ionic triangular lattices confined to cylindrical fibers.

We study the optimal packing of triangular ionic lattices on the surface of nanofibers. We compute the favored orientation of the lattice with respect to the axis of the cylindrical fiber, and we determine the effects of the surface curvature. Electrostatic interactions prefer chiral arrangements only for special families of lattices that depend on the fiber diameter. However, there are families of lattices that energetically promote achiral configurations. Besides the long-range Coulomb interactions we consider the behavior of short-range elastic forces, represented by interconnected springs between neighboring ions. In this case a different family of achiral lattices is always preferred. We also show that varying the stoichiometric composition of charges, as well as including higher-order curvature effects, does not significantly modify such a scenario.

Topological classification of RNA structures.

We present a novel topological classification of RNA secondary structures with pseudoknots. It is based on the topological genus of the circular diagram associated to the RNA base-pair structure. The genus is a positive integer number whose value quantifies the topological complexity of the folded RNA structure. In such a representation, planar diagrams correspond to pure RNA secondary structures and have zero genus, whereas non-planar diagrams correspond to pseudoknotted structures and have higher genus. The topological genus allows for the definition of topological folding motifs, similar in spirit to those introduced and commonly used in protein folding. We analyze real RNA structures from the databases Worldwide Protein Data Bank and Pseudobase and classify them according to their topological genus. For simplicity, we limit our analysis by considering only Watson-Crick complementary base pairs and G-U wobble base pairs. We compare the results of our statistical survey with existing theoretical and numerical models. We also discuss possible applications of this classification and show how it can be used for identifying new RNA structural motifs.

Spontaneous chirality via long-range electrostatic forces.

We consider a model for periodic patterns of charges constrained over a cylindrical surface. In particular we focus on patterns of chiral helices, achiral rings, or vertical lamellae, with the constraint of global electroneutrality. We study the dependence of the patterns' size and pitch angle on the radius of the cylinder and salt concentration. We obtain a phase diagram by using numerical and analytic techniques. For pure Coulomb interactions, we find a ring phase for small radii and a chiral helical phase for large radii. At a critical salt concentration, the characteristic domain size diverges, resulting in an achiral macroscopic phase-segregated structure. We discuss possible consequences and generalizations of our model.

Enumeration of RNA structures by matrix models.

We enumerate the number of RNA contact structures according to their genus, i.e., the topological character of their pseudoknots. By using a recently proposed matrix model formulation for the RNA folding problem, we obtain exact results for the simple case of an RNA molecule with an infinitely flexible backbone, in which any arbitrary pair of bases is allowed. We analyze the distribution of the genus of pseudoknots as a function of the total number of nucleotides along the phosphate-sugar backbone.