[Paper Review] Tree Orbits under Permutation Group Action: Algorithm, Enumeration and Application to Viral Assembly

Miklós Bóna, Meera Sitharam+1 more|ArXiv.org|Jun 1, 2009

Bacteriophages and microbial interactions23 references3 citations

TL;DR

This paper develops a combinatorial and algorithmic framework to enumerate and analyze assembly trees of icosahedral viral capsids under the action of the icosahedral group. It presents a linear-time algorithm to compute the stabilizer of a tree and derives a closed-form formula for counting tree orbits by size, enabling the calculation of assembly pathway probabilities in viral self-assembly models.

ABSTRACT

This paper uses combinatorics and group theory to answer questions about the assembly of icosahedral viral shells. Although the geometric structure of the capsid (shell) is fairly well understood in terms of its constituent subunits, the assembly process is not. For the purpose of this paper, the capsid is modeled by a polyhedron whose facets represent the monomers. The assembly process is modeled by a rooted tree, the leaves representing the facets of the polyhedron, the root representing the assembled polyhedron, and the internal vertices representing intermediate stages of assembly (subsets of facets). Besides its virological motivation, the enumeration of orbits of trees under the action of a finite group is of independent mathematical interest. If $G$ is a finite group acting on a finite set $X$, then there is a natural induced action of $G$ on the set $\mathcal{T}_X$ of trees whose leaves are bijectively labeled by the elements of $X$. If $G$ acts simply on $X$, then $|X| := |X_n| = n \cdot |G|$, where $n$ is the number of $G$-orbits in $X$. The basic combinatorial results in this paper are (1) a formula for the number of orbits of each size in the action of $G$ on $\mathcal{T}_{X_n}$, for every $n$, and (2) a simple algorithm to find the stabilizer of a tree $τ\in \mathcal{T}_X$ in $G$ that runs in linear time and does not need memory in addition to its input tree.

Motivation & Objective

• To model viral capsid assembly as rooted trees with monomers as leaves and subassemblies as internal nodes.
• To understand the probability distribution of assembly pathways under icosahedral symmetry using group action on trees.
• To develop an efficient algorithm for computing the stabilizer of a tree under group action.
• To enumerate tree orbits by size under the action of the icosahedral group, particularly for T=1 capsids with 60 facets.
• To apply the results to estimate the likelihood of specific viral assembly pathways based on geometric and symmetry factors.

Proposed method

• Models viral capsid assembly as rooted trees where leaves are monomers (facets), internal nodes are subassemblies, and the root is the complete capsid.
• Applies the icosahedral group $G_{60}$ to act on the set of labeled trees via permutation of leaf labels, inducing an action on the tree space $\mathcal{T}_X$.
• Uses Möbius inversion on the subgroup lattice of $G_{60}$ to compute the number of trees fixed by each subgroup $G_i$, enabling orbit enumeration.
• Derives a generating function $f_{G_i}(x)$ for trees fixed by subgroup $G_i$, using the structure of the subgroup lattice and Möbius function values.
• Employs a linear-time algorithm to compute the stabilizer of a given tree $\tau$ in $G$, based on permutation group theory and without requiring additional memory.
• Applies Theorem 2 and Theorem 3 to compute the number of trees with stabilizer exactly $G_i$, yielding the number of orbits of size $i$.

Experimental results

Research questions

• RQ1 How many distinct assembly pathways (tree orbits) exist under the icosahedral group action for a T=1 capsid with 60 monomers?
• RQ2 What is the probability distribution of assembly pathways, given their orbit sizes under the group action?
• RQ3 Can an efficient algorithm compute the stabilizer of a given tree under the icosahedral group action?
• RQ4 How do geometric stability and symmetry factors influence the likelihood of specific assembly pathways?
• RQ5 What is the combinatorial structure of trees that are fixed only by specific subgroups of the icosahedral group?

Key findings

• For the $T=1$ case with $|X|=60$, the number of assembly trees with stabilizer exactly $G_1$ (trivial group) is $\overline{t}_{60}(G_1) = 1924465510132437394720184730922187571120346754532$.
• The number of trees with stabilizer $G_{60}$ (full icosahedral group) is $\overline{t}_1(G_{60}) = 204$, corresponding to the most symmetric pathways.
• The number of trees with stabilizer $G_2$ (index 30) is $\overline{t}_{30}(G_2) = 1670856367100496379411587456529324583988755126499875584$.
• The number of trees with stabilizer $G_3$ (index 20) is $\overline{t}_{20}(G_3) = 10087157294451731428720995944759704$.
• The number of trees with stabilizer $G_5$ (index 12) is $\overline{t}_{12}(G_5) = 20540071766413107840$.
• Despite having only 204 symmetric pathways ($G_{60}$-invariant), the total number of asymmetric pathways ($G_1$-invariant) exceeds $10^{99}$, illustrating the vast diversity of possible assembly routes.