FIGURE 7. Ambiguity in data interpretation and conditional restraints.

a, The ambiguity for a protein interaction between proteins of green and yellow types is illustrated. The ambiguity results from the presence of multiple copies of the same protein in the same or neighbouring symmetry unit. In our NPC calculations, both neighbouring half-spokes on the cytoplasmic and nucleoplasmic sides are considered, for a total of four neighbouring half-spokes (not shown). b, The conditional restraint is illustrated by an example of a composite of four protein types (yellow, blue, red, green), derived from an assembly containing a single copy of the yellow, blue, and red protein and two copies of the green protein; proteins are represented by a single bead (blue protein), a pair of beads (green and red proteins), and a string of three beads (yellow protein) (right panel). This composite implies that at least three of the following six possible types of interaction must occur: blue–red, blue–yellow, blue–green, red–green, red–yellow and yellow–green. In addition, (1) the three selected interactions must form a 'spanning tree' of the 'composite graph' (defined below); (2) each type of interaction can involve either copy of the green protein (in general, all alternatives must be considered as illustrated in a); and (3) each protein can interact through any of its beads. These considerations can be encoded through a tree-like evaluation of the conditional restraint. At the top level, all optional bead–bead interactions between all protein copies are clustered by protein types. Each alternative bead interaction is restrained by a harmonic upper bound on the distance between the beads; these are 'optional restraints', because only a subset is selected for contribution to the final value of the conditional restraint. Next, a 'rank-and-select' operator (O_RS) selects only the least violated optional restraint from each interaction type, resulting in six restraints (thick red line) at the middle level of the tree. Finally, the minimal spanning tree operator (O_MST) finds the combination of three restraints that are most consistent with the composite data (thick red line); here the edge weights in the minimal spanning tree (defined below) correspond to the restraint values given the current assembly structure. The column on the right shows a structural interpretation of the composite with proteins represented by their coloured beads and alternative interactions indicated by edges between them. The composite graph (shown on the left) is a fully connected graph that consists of nodes for all identified protein types and edges for all pairwise interactions between protein types; in the context of the conditional restraint, the edge weights correspond to the restraint values. Five of the sixteen possible spanning trees are also shown. A spanning tree is a graph with the smallest possible number of edges that connect all nodes. The minimal spanning tree is the spanning tree with the minimal sum of edge weights. This restraint evaluation process is executed at each optimization step based on the current configuration, thus resulting in possibly different subsets of selected optional restraints at each step.