We propose a simple and general model accounting for the dependence of the viscosity of a hard sphere suspension at arbitrary volume fractions. The model constitutes a continuummedium description based on a recursive-differential method that assumes a hierarchy of relaxation times. Geometrical information of the system is introduced through an effective volume fraction that approaches the usual filling fraction at low concentrations and becomes one at maximum packing. The agreement of our expression for the viscosity with experiments at low-and high-shear rates and in the high-frequency limit is remarkable for all volume fractions.

Many physical systems including lattices near structural phase transitions, glasses, jammed solids and biopolymer gels have coordination numbers placing them at the edge of mechanical instability. Their properties are determined by an interplay between soft mechanical modes and thermal fluctuations. Here we report our investigation of the mechanical instability in a lattice model at finite temperature T. The model we used is a square lattice with a f 4 potential between next-nearest-neighbour sites, whose quadratic coefficient k can be tuned from positive to negative. Using analytical techniques and simulations, we obtain a phase diagram characterizing a first-order transition between the square and the rhombic phase and different regimes of elasticity, as well as an 'order-bydisorder' effect that favours the rhombic over other zigzagging configurations. We expect our study to provide a framework for the investigation of finite-T mechanical and phase behaviour of other systems with a large number of floppy modes.

We propose an improved effective-medium theory to obtain the concentration dependence of the viscosity of particle suspensions at arbitrary volume fractions. Our methodology can be applied, in principle, to any particle shape as long as the intrinsic viscosity is known in the dilute limit and the particles are not too elongated. The procedure allows to construct a continuum-medium model in which correlations between the particles are introduced through an effective volume fraction. We have tested the procedure using spheres, ellipsoids, cylinders, dumbells, and other complex shapes. In the case of hard spherical particles, our expression improves considerably previous models like the widely used Krieger-Dougherty relation. The final expressions obtained for the viscosity scale with the effective volume fraction and show remarkable agreement with experiments and numerical simulations at a large variety of situations.

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CcmK proteins are major constituents of icosahedral shells of β-carboxysomes, a bacterial microcompartment that plays a key role for CO2 fixation in nature. Supported by the characterization of bidimensional (2D) layers of packed CcmK hexamers in crystal and electron microscopy structures, CcmK are assumed to be the major components of icosahedral flat facets. Here, we reassessed the validity of this model by studying CcmK isoforms from Synechocystis sp. PCC6803. Native mass spectrometry studies confirmed that CcmK are hexamers in solution. Interestingly, potential pre-assembled intermediates were also detected with CcmK2. Atomic-force microscopy (AFM) imaging under quasi-physiological conditions confirmed the formation of canonical flat sheets with CcmK4. Conversely, CcmK2 formed both canonical and striped-patterned patches, while CcmK1 assembled into remarkable supra-hexameric curved honeycomb-like mosaics. Mutational studies ascribed the propensity of CcmK1 to form round assemblies to a combination of two features shared by at least one CcmK isoform in most β-cyanobacteria: a displacement of an α helical portion towards the hexamer edge, where a potential phosphate binding funnel forms between packed hexamers, and the presence of a short C-terminal extension in CcmK1. All-atom molecular dynamics supported a contribution of phosphate molecules sandwiched between hexamers to bend CcmK1 assemblies. Formation of supra-hexameric curved structures could be reproduced in coarse-grained simulations, provided that adhesion forces to the support were weak. Apart from uncovering unprecedented CcmK self-assembly features, our data suggest the possibility that transitions between curved and flat assemblies, following cargo maturation, could be important for the biogenesis of β-carboxysomes, possibly also of other BMC.