Chromonic liquid crystals and packing configurations of bacteriophage viruses

Abstract

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’.

Helical organization of DNA-like liquid crystal filaments in cylindrical viral capsids

“…Table 1 lists the value of m0 for a sample of four viruses. For instance, for T4, we estimate m0=1/false(πfalse(d0/2false)2false)=0.221 nm−2 (d0≈2.4 nm), and taking T=300 K gives K3=5×10−11 J normalm−1.In [27], we take guidance from the theory of Onsager for lyotropic liquid crystals, to obtain expressions for the isotropic modulus ν and the surface tension σ, and assume that they are functions of the (DNA) molar concentration c [45]. We adopt the expressions ν=ν0false(cfalse)KBTR231emand1emσ=σ0false(cfalse)KBTLpd0.Since, to our knowledge, no molecular theory is a...…”

“…Since, to our knowledge, no molecular theory is available to determine the dimensionless parameters ν 0 and σ 0 , and, likewise, we do not have an expression for K 2 either, we proceed to estimate these three quantities from the data shown in the table. (The analogous approach followed in [27], and taking the capsid to be a sphere with the DNA arranged in concentric circles, gives ν 0 = 23 and σ 0 = 0.388.) Prior to estimating ν 0 and σ 0 , and taking into account that for DNA α > 1 holds, the stability properties listed in §3d(i) indicate that the solution R 1 in the graphs shown in figure 3 lies either on the concentric circle branch or on the monotonically decreasing portion corresponding to α = 10.…”

“…For the class of vector fields bold-italicn that we consider, including the cross-sectional energy amounts to replacing the constants K2 and K3 by effective values that account for the shear and bulk elastic modulus, but it does not otherwise affect the analysis presented here. This simplifying approach may have to be reconsidered if the goal is to predict the osmotic pressure in the capsid [27].…”

“…In this paper, we consider the previously introduced continuum model that endows condensed DNA with a structure locally consisting of curvilinear segments (representing the DNA centre axis) together with their orthogonal cross sections [27]. Assigning these segments an effective diameter yields a plane hexagonal crystal structure on the intersecting planes.…”

“…Accordingly, the energy consists of the nematic Oseen-Frank contribution penalizing changes of direction of the vector field (and so that of the DNA axis) and the isotropic energy of the core. Moreover, for the class of helical vector fields (3.3), the former also accounts for the elastic energy associated with the disruption of the hexagonal cross-sectional structure (remark 2.2 and [27]). Furthermore, since images reveal a sharp transition between the ordered and disordered regions of the capsid, we include a surface energy term that tends to minimize the surface area of the interface.…”