Knotted DNA Configurations in Bacteriophage Capsids: A Liquid Crystal Theory Approach

Abstract. Bacteriophages, viruses that infect bacteria, store their micron long DNA inside an icosahedral capsid with a typical diameter of 40 nm to 100 nm. Consistent with experimental observations, such confinement conditions induce an arrangement of DNA that corresponds to a hexagonal chromonic liquid-crystalline phase, and increase the topological complexity of the genome in the form of knots. A mathematical model that implements a chromonic liquid-crystalline phase and that captures the changes in topology has been lacking. We adopt a mathematical model that represents the viral DNA as a pair of a vector field and a line. The vector field is a minimizer of the total Oseen–Frank energy for nematic liquid crystals under chromonic constraints, while the line is identified with the tangent to the field at selected locations, representing the central axis of the DNA molecule. The fact that the Oseen–Frank functional assigns infinite energy to topological defects (point defects in two dimensions and line defects in three dimensions) precludes the presence of singularities and, in particular, of knot structures. To overcome this limitation, we develop an algorithm that starts from the optimal vector field and an associated helical curve and introduces knots through stochastic perturbations driven by splay and twist deformations, modeled via a Langevin system. We conclude by comparing the resulting knot distributions with experimentally observed data and by interpreting the results in light of previously published measurements. Overall, the proposed approach highlights the interplay of modeling, analysis, and computation in the study of viral DNA organization within capsids.