Tactoids are droplets of a nematic phase that under suitable conditions form in dispersions of elongated colloidal particles. We theoretically study the shape and the director-field configuration of such droplets for the case where a planar anchoring of the director field to the interface is favored. A minimum of four regimes can be identified in which the droplets have a different structure. Large droplets tend to be nearly spherical with a director field that is bipolar if the surface tension is strongly anisotropic and homogeneous if this is not so. Small droplets can become very elongated and spindlelike if the surface tension is sufficiently anisotropic. Depending on the anchoring strength, the director field is then either homogeneous or bipolar. We find that the more elongated the tactoid, the more strongly it resists the crossing over from a homogeneous to a bipolar structure. This should have implications for the nucleation rate of the nematic phase. Our calculations qualitatively describe the size dependence of the aspect ratio of tactoids found in recent experiments.

We present a theory of the dependence on sequence of the three-dimensional size of large single-stranded (ss) RNA molecules. The work is motivated by the fact that the genomes of many viruses are large ssRNA molecules-often several thousand nucleotides long-and that these RNAs are spontaneously packaged into small rigid protein shells. We argue that there has been evolutionary pressure for the genome to have overall spatial properties-including an appropriate radius of gyration, R g-that facilitate this assembly process. For an arbitrary RNA sequence, we introduce the (thermal) average maximum ladder distance (͗MLD͘) and use it as a measure of the ''extendedness'' of the RNA secondary structure. The ͗MLD͘ values of viral ssRNAs that package into capsids of fixed size are shown to be consistently smaller than those for randomly permuted sequences of the same length and base composition, and also smaller than those of natural ssRNAs that are not under evolutionary pressure to have a compact native form. By mapping these secondary structures onto a linear polymer model and by using ͗MLD͘ as a measure of effective contour length, we predict the R g values of viral ssRNAs are smaller than those of nonviral sequences. More generally, we predict the average ͗MLD͘ values of large nonviral ssRNAs scale as N 0.67؎0.01 , where N is the number of nucleotides, and that their R g values vary as ͗MLD͘ 0.5 in an ideal solvent, and hence as N 0.34 . An alternative analysis, which explicitly includes all branches, is introduced and shown to yield consistent results.branched polymer ͉ ladder distance ͉ radius of gyration ͉ secondary structure ͉ viral RNA

We theoretically investigate the director field inside spindle-shaped nematic droplets, known as tactoids. Tactoids typically form in dispersions of rod-like colloidal particles. By optimising the bulk elastic and surface energies, we find that the director field crosses over smoothly from a homogeneous to a bipolar configuration with increasing droplet size, in a process that we postulate to involve two virtual point defects that move in from infinity towards the poles on the surface of the droplet. Our calculations show that these hypothesised virtual defects become true surface point defects or boojums only in the limit of infinite droplet volume, and that the more elongated the droplets are, the larger their volume has to be before a uniform director field distorts so as to become discernibly bipolar. The theory agrees well with available experimental data on the size dependence of the aspect ratio of tactoids.

We theoretically investigate under what conditions the director field in a spindle-shaped nematic droplet or tactoid obtains a twisted, parity-broken structure. By minimizing the sum of the bulk elastic and surface energies, we show that a twisted director field is stable if the twist and bend elastic constants are small enough compared to the splay elastic constant, but only if the droplet volume is larger than some minimum value. We furthermore show that the transition from an untwisted to a twisted director-field structure is a sharp function of the various control parameters. We predict that suspensions of rigid, rod-like particles cannot support droplets with a parity broken structure, whereas they could possibly occur in those of semi-flexible, worm-like particles.

The evolution of the contact zone between pure surfactant and solvent has been studied by mesoscale simulation. It is found that mesophase formation becomes diffusion controlled and follows the equilibrium phase diagram adiabatically almost as soon as individual mesophases can be identified, corresponding to times in real systems of order 10 µs.When pure surfactant comes into contact with water, mesophases appear at the interface. This is an important process not only for the practical use of surfactants, but also from the point of view of fundamental surfactant phase science. Indeed, contact 'flooding' or penetration scan experiments can yield quantitative information on mesophases as a function of composition [1]. The phenomenon is invariably diffusion controlled in the sense that the widths of the mesophases follow t 1/2 growth laws, and adiabatic in the sense that the local composition determines the mesophase boundaries according to the equilibrium phase diagram [2,3,4]. But penetration scan experiments are restricted to observation times of minutes to hours: what happens on time scales shorter than this? How early does diffusion control set in, and how soon can one expect the mesophase boundaries to track the local composition adiabatically? Such questions are not just of scientific interest since time scales of seconds or less are important in modern processing and the everyday use of surfactants, for instance determining how rapidly washing powder dissolves. Experimentally, this regime is very difficult to access because of the short time scales and the relatively small amounts of mesophase involved. To probe these questions therefore, we have therefore undertaken novel mesoscale simulations of surfactant dissolution. We find adiabatic diffusion control is established remarkably rapidly in our model, on time scales in which only a few repeat units of the growing mesophases have appeared, corresponding to times in the real systems of order 10 µs.The model we have used is a 'minimalist' particlebased model of a binary surfactant / water mixture, based on the dissipative particle dynamics (DPD) method [5,6]. In DPD, the particles are soft spheres, interacting with pairwise soft potentials of the form U = 1 2 A(1 − r/r c ) 2 (r < r c ) where r is the particle separation, r c is the range of the interaction, and A the amplitude. In the model we have three species of particles: A, B and C. The A and B particles are bound together in pairs as dimers with a fixed separation r d , and represent the surfactants. The C particles are monomers representing the solvent (water). The different species are distinguished by their interaction amplitudes, and the trick is to find a set of amplitudes which recover suitable phase behaviour. Following earlier work [6], we use A AA = A BB = A CC = 25, A AB = 30, A AC = 0, A BC = 50 and r d = 0.5 which gives phase diagram features lying in a convenient temperature range around k B T ∼ 1 (we fix units by choosing m = r c = 1 where m is the mass of the particles). Fig. 1 shows the...

A theory is set up of spherical proteins interacting by screened electrostatics and constant adhesion, in which the effective adhesion parameter is optimized by a variational principle for the free energy. An analytical approach to the second virial coefficient is first outlined by balancing the repulsive electrostatics against part of the bare adhesion. A theory similar in spirit is developed at nonzero concentrations by assuming an appropriate Baxter model as the reference state. The first-order term in a functional expansion of the free energy is set equal to zero which determines the effective adhesion as a function of salt and protein concentrations. The resulting theory is shown to have fairly good predictive power for the ionic-strength dependence of both the second virial coefficient and the osmotic pressure or compressibility of lysozyme up to about 0.2 volume fraction.

We show on general theoretical grounds that the two ends of single-stranded (ss) RNA molecules (consisting of roughly equal proportions of A, C, G and U) are necessarily close together, largely independent of their length and sequence. This is demonstrated to be a direct consequence of two generic properties of the equilibrium secondary structures, namely that the average proportion of bases in pairs is ∼60% and that the average duplex length is ∼4. Based on mfold and Vienna computations on large numbers of ssRNAs of various lengths (1000–10 000 nt) and sequences (both random and biological), we find that the 5′–3′ distance—defined as the sum of H-bond and covalent (ss) links separating the ends of the RNA chain—is small, averaging 15–20 for each set of viral sequences tested. For random sequences this distance is ∼12, consistent with the theory. We discuss the relevance of these results to evolved sequence complementarity and specific protein binding effects that are known to be important for keeping the two ends of viral and messenger RNAs in close proximity. Finally we speculate on how our conclusions imply indistinguishability in size and shape of equilibrated forms of linear and covalently circularized ssRNA molecules.

Under conditions of low ionic strength and a pH ranging between about 3.7 and 5.0, solutions of purified coat proteins of cowpea chlorotic mottle virus (CCMV) form spherical multishell structures in the absence of viral RNA. The outer surfaces of the shells in these structures are negatively charged, whereas the inner surfaces are positively charged due to a disordered cationic N-terminal domain of the capsid protein, the arginine-rich RNA-binding motif that protrudes into the interior. We show that the main forces stabilizing these multishells are counterion release combined with a lower charge density in the RNA-binding motif region of the outer shells due to their larger radii of curvature, arguing that these compensate for the outer shells not being able to adopt the smaller, optimal, radius of curvature of the inner shell. This explains why the structures are only stable at low ionic strengths at pHs for which the outer surface is negatively charged and why the larger outer shells are not observed separately in solution. We show how to calculate the free energy of shells of nonoptimal radius of curvature from the elastic properties of the native shell. The spacing between shells is determined mainly by the entropic elasticity of the RNA-binding motifs. Although we focus on CCMV multishells, we also predict the solution conditions under which multishells formed by CCMV coat protein mutants with a lower RNA-binding motif charge are stable, and we examine other viruses as well. We conclude that at a given surface charge density, the boundaries separating regions of stable multishells with different numbers of shells shift to lower ionic strengths upon either increasing the length of the RNAbinding motif, increasing the stiffness of the shells, or decreasing the charge per RNA-binding motif.