Fig 1.
The model system—bacteriophage MS2.
(A) The viral capsid is formed from 60 asymmetric and 29 symmetric copies of the CP dimers, with one MP that takes the place of a symmetric dimer (PDBID 2MS2). The genomic RNA is organized inside the particles in two shells, with the outer shell adopting the shape of a polyhedral cage in icosahedrally-averaged reconstructions. (B) Depiction of the polyhedral cage, showing long (purple) and short (orange) PS-PS RNA connections. (C) Asymmetrically averaged tomogram of bacteriophage MS2 bound to its receptor, the bacterial F-pilus. The portion of the electron density corresponding to the CP shell (and bacterial pilus) is shown in blue; green depicts the density for genomic RNA (and presumably some elements of the MP), which forms the basis for the analysis described in this study. The RNA density forms a shell that is intimately associated with the inside surface of the capsid. (D) A planar representation of protein container and polyhedral RNA organization, showing the relative positions of the 60 polyhedral vertices (PS positions, indicated as yellow circles) in contact with the 60 asymmetric CP dimers.
Fig 2.
Classification of polyhedral edges as occupied and non-occupied.
A comparison of the density profiles of the sampled long edge connections. The mean of a fitted normal distribution (y-axis) is scattered with a skewness parameter (x-axis). Connections with negative skew are disregarded as no statement about occupancy can be deduced in this case. From the remainder, two groups of four and five connections are identified as occupied (in the green circle) and non-occupied (red circle), respectively. These are used as constraints in the analysis.
Fig 3.
Constraints on the RNA organization consistent with the tomogram.
Each possible RNA organization is characterized by which long edges (Fig. 1B, purple edges) are occupied in the polyhedral shell of the icosahedrally-averaged density. Long connections are labelled by the numbers of the five-fold vertices (Fig. 1D) they connect (x•y connecting five-fold vertices x and y). Constraints imposed in the analysis are indicated in the first row, with green indicating an occupied edge, and red an unoccupied edge. The five paths meeting these constraints are characterized according to occupied and non-occupied edges. The last row shows edges shared by all five paths.
Fig 4.
Symmetry averaging identifies Path 4 as the correct solution.
C5-averaged densities in 1-D projection for tomographic data and the path solutions listed in Fig. 3 are compared. The vertical axis shows the radial distance from the centre of the capsid in angstrom, and the horizontal axis corresponds to the C5-averaged density at that radial distance in arbitrary units; density profiles for tomogram and path solutions are normalized by equalizing the maximum densities. Density profiles are shown for: (A) the average of all possible 40,678 Hamiltonian paths; (B) the average of all paths consistent with RNA interaction with the MP; (C-G) the density profiles for the five paths in Fig. 3 individually; (H) the C5 cryo-EM reconstruction from the tomogram, adapted from [20]. Path 4 (cf. Fig. 5A), identical to Path 3 (Fig. 5B) from a geometric point of view but positioned differently within the density with respect to MP, provides the closest fit with the cryo-EM data.
Fig 5.
Hamiltonian path solution identified by the method.
(A) The best match with the C5 averaged data (Path 4) starts and finishes at vertex 5 adjacent to the MP (cyan). Following the colouring convention in Fig. 3, red dashed lines show unoccupied and green lines occupied constraints; other occupied connections implied by our analysis are shown in black. The position of TR, the strongest PS, is denoted in yellow [15]; heterodimers are coloured in green-blue and homodimers in pink. (B) An alternative embedding of the same (geometric) path with a different orientation relative to MP. The path (Path 3) starts and finishes at vertex 9; hence the occupation of the connections differs from Path 4 in (A), even though the overall geometry of the path is the same.