Molecular and mathematical foundations of the local-rule theory of icosahedral virus shell assembly: what enforces the rules, and do local rules exist for every triangulation number?

Berger, Shor, Tucker-Kellogg and King (PNAS 1994) proposed that icosahedral virus shells self-assemble under local rules: coat proteins adopt a small number of discrete conformations, and each conformation follows pairwise binding preferences (allowed partner conformations with approximate interaction angles, lengths and torsions) that, followed in any order, close the correct shell. Simulations closed shells for a T=7 bacteriophage (P22, using seven or a reduced set of four conformations with a 'disallowed-hexagon' rule) and for the all-pentamer, quasi-equivalence-violating capsid of simian virus 40 (SV40). The paper left the theory's foundations open: the true number of distinct P22 coat-protein conformations, the molecular mechanism (trimer energetics or scaffolding proteins) that enforces the disallowed-hexagon rule in the reduced set, the role of SV40 C-terminal arms in enforcing local rules, the conditions for assembly initiation and the role of P22 scaffolding protein, how local-rule parameters relate to spherical-versus-polyhedral shell shape, whether local rules could be exploited to design assembly inhibitors, and whether local-rule constructions -- especially reduced-conformation sets -- exist for every allowed Caspar-Klug triangulation number $T$. Three decades of follow-up (ambiguous-rule and trimer-constraint simulations, viral tiling theory, local rules for non-quasi-equivalent capsids, atomic-resolution P22 structures, polyomavirus and SV40 assembly studies, demonstrated assembly-directed antivirals for hepatitis B, and geometric analyses linking subunit dihedral angles to capsid size and buckling) have advanced several components but not closed the core: the molecular identification of the conformational states and rule-enforcement interactions in specific viruses, and a rigorous constructive theory of local rules for all $T$ numbers, both remain open.

Background

Most spherical viruses build their protein shells (capsids) with icosahedral symmetry from many copies of one coat protein. The Caspar-Klug quasi-equivalence theory (Caspar and Klug, 1962) classifies such shells by a triangulation number $T = f^2(h^2 + hk + k^2)$ for nonnegative integers $f$, $h$, $k$: the shell contains $60T$ subunits arranged as 12 pentamers and $10(T-1)$ hexamers, with subunits in $T$ distinct (quasi-equivalent) local environments. Bacteriophage P22 is a $T = 7$ system (420 coat subunits) whose empty precursor shell, the procapsid, is assembled with the help of an internal scaffolding protein that exits during maturation; cryo-electron microscopy (Prasad et al., 1993) showed that all seven quasi-equivalent subunits of the T=7 lattice change conformation as the spherical procapsid expands into the polyhedral mature capsid. Simian virus 40 (SV40), a polyomavirus, violates quasi-equivalence altogether: its capsid is built from 72 pentamers of the coat protein VP1, some of which contact six neighbours, a pattern that Caspar-Klug theory disallows; the crystal structure (Liddington et al., 1991) showed that flexible C-terminal arms of VP1 extend from each pentamer into its neighbours, tying the shell together. Berger, Shor, Tucker-Kellogg and King (1994) introduced the local-rule theory to explain how such shells can assemble without preformed building blocks or any global blueprint: each subunit switches among a small set of discrete conformations, and a rule set specifies, per conformation, which conformations it may bind and the approximate interaction angles, interaction lengths and torsional angles of each allowed pair. Computer simulations showed that following these local rules assembles correct closed shells (T=3 and T=7, and an SV40-like all-pentamer shell with six conformations whose inter-pentamer contacts are mediated by C-terminal arms), that assembly tolerates roughly 8 percent deviations in angles and lengths before failing, and that seeding growth with a hexamer instead of a pentamer produces the non-closing 'spiralling' malformations actually observed in P22. For P22 the paper gave a seven-conformation rule set (one conformation per quasi-equivalent position) and an alternative four-conformation set, in which hexamers are symmetric under 180-degree rotation and a 'disallowed-hexagon' rule forbids hexamers containing three subunits of the fourth conformation. The unresolved issues the paper itself identified -- the subject of this problem -- concern the molecular reality and enforcement of these rules (what the assembling coat proteins actually do, and what interactions, including scaffolding proteins and inter-pentamer arms, enforce rule selection and initiation) and the mathematical generality of the construction (whether local-rule sets, especially reduced-conformation sets, exist for every allowed $T$).

Problem Statement

Establish the molecular and mathematical foundations of the local-rule theory of icosahedral virus shell assembly, as posed by the open issues of Berger, Shor, Tucker-Kellogg and King (1994). Concretely: (1) Molecular basis of the rules. Determine the actual conformational states that the assembling coat protein adopts -- for bacteriophage P22, whether the seven quasi-equivalent conformations observed in the procapsid are genuinely distinct or whether assembly uses a reduced set (such as the four-conformation set, in which hexamers are symmetric under 180-degree rotation), and identify the molecular interaction, for example energetics of trimer formation or interactions with scaffolding protein, that enforces the disallowed-hexagon rule of the reduced set and thereby selects the T=7 geometry; determine the role of SV40 VP1 C-terminal (inter-pentamer) and N-terminal (intra-pentamer) arms in enforcing the local rules of the all-pentamer, quasi-equivalence-violating capsid; and determine the conditions for assembly initiation and the role of scaffolding protein in P22. (2) Mathematical generality. Decide whether, for every allowed Caspar-Klug triangulation number $T = f^2(h^2 + hk + k^2)$, there exists a constructive set of local rules (per-conformation binding partners with approximate interaction angles, lengths and torsions) such that following the rules in any order assembles the correct closed icosahedral shell of $60T$ subunits and nothing else, and in particular whether reduced-conformation rule sets using fewer than $T$ distinct conformations exist for all $T$. For definiteness, a rule set is understood operationally as in Berger et al. (1994): each rule fixes, per conformation, the allowed partner conformations and the approximate interaction angles, lengths and torsions together with their tolerance margins; an actual contact matches a rule when its geometry falls within those margins; 'in any order' asserts that the correct shell is assembled regardless of the order in which the allowed bindings are followed; and 'nothing else' asserts that no other closed shell is assembled. (3) Consequences of the rules. Characterize how local-rule parameters (binding angles and lengths, and their tolerances) map to shell shape variation between spherical and polyhedral forms, and whether local rules can be deliberately subverted to design assembly inhibitors. The molecular and shape determinations are established by argument from the cited structural and biochemical evidence, with no new experimental data required, and the mathematical statements by explicit construction or obstruction; partial answers that resolve only one of the molecular systems, or only the reduced-construction question for particular $T$ values, leave the problem open.

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