A theory of the symmetries of filamentous bacteriophages

A mathematical model is presented which explains the symmetries observed for the protein coats of filamentous bacterial viruses. Three viruses (Ff, IKe, and If1) all have five-start helices with rotation angles of 36 degrees and axial translations of 16 A (Type I symmetry), and three other viruses (Pf1, Xf, and Pf3) all have one-start helices with rotation angles of approximately equal to 67 degrees and translations of approximately 3 A (Type II symmetry). The coat protein subunits in each group diverge from each other in amino acid sequence, and Type II viruses differ dramatically in DNA structure. Regardless of the differences, both Type I and Type II symmetry can be understood as direct, natural consequences of the close-packing of alpha-helical protein subunits. In our treatment, an alpha-helical subunit is modeled as consisting of two interconnected, flexible tubular segments that follow helical paths around the DNA, one in an inner layer and the other in an outer layer. The mathematical model is a set of algebraic equations describing the disposition of the flexible segments. Solutions are described by newly introduced symmetry indices and other parameters. An exhaustive survey over the range of indices has produced a library of all structures that are geometrically feasible within our modeling scheme. Solutions which correspond in their rotation angles to Type I and Type II viruses occur over large ranges of the parameter space. A few solutions with other symmetries are also allowed, and viruses with these symmetries may exist in nature. One solution to the set of equations, obtained without any recourse to the x-ray data, yields a calculated x-ray diffraction pattern for Pf1 which compares reasonably with experimental patterns. The close-packing geometry we have used helps explain the near constant linear mass density of known filamentous phages. Helicoid, rigid cylinder, and maximum entropy structure models proposed by others for Pf1 are reconciled with the flexible tube models and with one another.