We study equilibrium configurations of hexagonal columnar liquid crystals in the context of characterizing packing structures of bacteriophage viruses in a protein capsid. These are viruses that infect bacteria and are currently the focus of intense research efforts, with the goal of finding new therapies for bacteria-resistant antibiotics. The energy that we propose consists of the Oseen–Frank free energy of nematic liquid crystals that penalizes bending of the columnar directions, in addition to the cross-sectional elastic energy accounting for distortions of the transverse hexagonal structure; we also consider the isotropic contribution of the core and the energy of the unknown interface between the outer ordered region of the capsid and the inner disordered core. The problem becomes of free boundary type, with constraints. We show that the concentric, azimuthal, spool-like configuration is the absolute minimizer. Moreover, we present examples of toroidal structures formed by DNA in free solution and compare them with the analogous ones occurring in experiments with other types of lyotropic liquid crystals, such as food dyes and additives. This article is part of the theme issue ‘Topics in mathematical design of complex materials’.

In vivo configurations of dsDNA of bacteriophage viruses in a capsid are known to form hexagonal chromonic liquid crystal phases. This article studies the liquid crystal ordering of viral dsDNA in an icosahedral capsid, combining the chromonic model with that of liquid crystals with variable degree of orientation. The scalar order parameter of the latter allows us to distinguish regions of the capsid with well-ordered DNA from the disordered central core. We employ a state-of-the-art numerical algorithm based on the finite element method to find equilibrium states of the encapsidated DNA and calculate the corresponding pressure. With a data-oriented parameter selection strategy, the method yields phase spaces of the pressure and the radius of the disordered core, in terms of relevant dimensionless parameters, rendering the proposed algorithm into a preliminary bacteriophage designing tool. The presence of the order parameter also has the unique role of allowing for non-smooth capsid domains as well as accounting for knot locations of the DNA. A IntroductionThe discovery of liquid crystal phases formed by DNA in free solution as well as inside viral capsids led to the chromonic denomination of a class of lyotropic liquid crystals. Different from liquid crystals consisting of elongated, rod-like molecules found, for instance, in display devices, chromonic liquid crystals consist of disk, plank-like molecules that form liquid crystal phases with varying concentration. The molecules stack to form A-1 arXiv:1902.09188v1 [cond-mat.soft] 25 Feb 2019 cylindrical aggregates, and at even higher concentration, the cylinders cluster so their center axes align in a hexagonal lattice, forming what is known as the hexagonal columnar liquid crystal phase. Experimental and theoretical studies acquired over the last 30 years [22,23,25,37,39,42,36] have shown that encapsidated DNA molecules form a columnar hexagonal liquid crystal phase. In particular, liquid crystalline phases in bacteriophages were first proposed in [16], with an explicit reference to hexagonal packing made in [28] and since then, consistent data have been accumulating [21,20,30,37].This article studies the liquid crystal ordering of viral dsDNA in a capsid with axial symmetry. We use advanced liquid crystal models and their state-of-the-art numerical implementation to determine the equilibrium structure of the DNA in the capsid and the forces that it sustains. The model is also aimed to be used as a tool for the design and prediction of viral encapsidated DNA in response to the renewed interest in bacteriophages in medicine and biotechnology. This goal guides our parameter selection strategy which is also a main component of the work. The proposed method delivers a parameter phase space for the pressure and the the size of the inner disordered core.Icosahedral bacteriophages consist of a protein capsid whose assembly is followed by the packing, by means of a molecular motor, of a single naked dsDNA molecule [41]. The DNA molecule inside the viral capsid is found...

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