PMC

Gauge points impose a set of geometric constraints at different radii and angular locations throughout the capsid and genome. Here, we present 4 of the 55 base point arrays [,,]. (top) Gauge point 2, located along the 5–3 great circle (orange), is created by (200 points) and (192 points). There are two sister arrays for every gauge point and these produce nearly identical point arrays, with only their base being different (outlined in bold), either or . Here, we see that if a protruding features is located at gauge point 2, then there must be another material boundary at 83% of that radius located at the same angular location as gauge point 5, though this point is not considered a gauge point because it is not the most radially distal. Then another boundary at the 2-fold axes (IDD) at 71% of the gauge point radius and so on. Most of the structures we have studied only consider the protein capsid; however, the genome would also be subject to these constraints. (bottom) Gauge point 19, located along the 5–2 great circle (purple), is created by (342 points) and (360 points). Here, we see that if a protruding feature is found at GP 19, then the next material boundary must be located at 91% of this radius, at the same angular location of GP 5. Then at 74% of this radius, there is a bulk constraint, 120 points in total; see .

(a) The spatial distributions of gauge points have icosahedral symmetry, i.e., they are arranged with 2-, 3- and 5-fold symmetry as shown. The gauge points all lie on the 15 icosahedral great circles which connect nearest neighbor symmetry axes [] and are colored as in . We refer to the arcs of the circles subtending the 2-, 3- and 5-fold axes as the 5-2 GC, 5-3 GC and 2-3 GC. The volume bounded between these arcs is known as the asymmetric unit (AU) and is a representative one-sixtieth section of the entire capsid (yellow). (b) The gauge points have been placed on a radially colored HBV (1qgt) capsid at their nearest possible distance from the protein surface. We will later see that the only admissible gauge points are the purple and orange elements sitting atop the protruding dimers. (c) There are 21 unique gauge points in the AU, see . The gauge points are formed from 2-, 3- and 5-fold translations of the icosahedron (ICO), Dodecahedron (DOD) and Icosadodecahedron (IDD). The gauge points along the 2-, 3- and 5-fold axes result solely from , and point arrays, respectively. The off-axis gauge points, e.g., GP:2-GP:5, each result from only nearly identical point arrays known as sister point arrays. For example, Gauge Point 4 (GP:4) is only created from and . In total 36 of the 55-point arrays have gauge points off-axis. There are no gauge points in the bulk regions not located on the great circles, however interior points of the point arrays can be located there.

(a) A pair of yb gauge defects, shown by the vertices, can be caused by a string of incorrect yb face outcomes on the dual lattice. The displayed gauge syndrome is equivalent to that shown in where the measurement errors have occurred on the faces that returned −1 outcomes. (b) Three gauge defects, colored gy, gb and yb, can be caused by an error string that branches at a red cell. The branching point indicates a stabilizer defect at a red cell. (c) Strings of incorrect face outcomes of colors rg and gb terminate at a yellow boundary.

Features of the point array fitting algorithm. Here we have docked the gauge points on HBV, a T = 4 virus with dimeric protrusions. We only consider the point arrays with gauge points nearest the most radially distal protruding features, e.g., the nearest gauge points to the protruding dimers are GP: 17, 18 and 19 (yellow circles) and 4 and 5 (blue circles), see . In this example, there are 3 gauge points that would have fallen through the capsid surface when docked, which were instead stopped at their minimum distance to the surface. We see that most points sit on the surface of the proteins, and some on the symmetry axes in between proteins. Points which are near more than one protein are weighted more in the RMSD, through protein multiplicity , see Equation (). High multiplicity always occurs on symmetry axes e.g., Gauge Point 1 (red) on the 5-fold axis would count 5 times. There are several points which would lead to a high prevalence, i.e., they do not intersect proteins over a large range of radial scaling, including Gauge Point 15 (blue) which is on a 2-fold axis and Gauge Point 3 (orange). In this example, these points would lead to a poor RMSD fit, as they are each several angstroms away from the surface.