# Symmetry in Sphere-Based Assembly Configuration Spaces

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## Abstract

**:**

## 1. Motivation

- A collection of k rigid molecular components belonging to a few types; a rigid component is specified as the set of positions of the centers of their constituent atoms, in a local coordinate system. In many cases, an atom could be the representation of the average position of a collection of atoms in an amino acid residue. Note that an assembly configuration is given by the positions and orientations of the entire set of k rigid molecular components in an assembly system, relative to one fixed component. Since each rigid molecular component has six degrees of freedom, a configuration is a point in $6(k-1)$ dimensional Euclidean space.
- The pairwise component of the potential energy function of the assembly system is specified as a sum of potential energy terms between pairs of constituent atoms i and j in two different rigid components of the assembly system. The weak interaction between the rigid molecular components is captured by this potential energy function. The pairwise potential energy terms are, in turn, specified using pairwise potential energy functions similar to so-called Lennard–Jones potentials and Morse potentials [1]. The potential energy is a function of the distance ${d}_{i,j}$ between i and j.
- A non-pairwise component of the potential energy function is in the form of global potential energy terms that capture the tethers between the rigid components within a monomer, as well as other global potential energy terms that implicitly represent the solvent (water or lipid bilayer membrane) effect [2,3,4]. These are independent of particular pairs of atoms.

- Input: the 3D descriptions of the rigid molecular components and their interactions (Section 2 describes how they are formally specified). Output: prediction of the final assembly structures and their likelihood.
- Input: as in the previous item, plus a 3D configuration of the final assembled structure. Output: prediction of those interactions that are crucial for the assembly process to terminate in the given input assembly configuration.
- Input: as in the previous item. Output: prediction of minimal alterations of the building blocks or interactions that would significantly increase the likelihood of the assembly process terminating in the given input assembly configuration.
- Input: as in the previous item; additionally, more than one choice of final assembly configuration. Output: prediction of key events, such as specific intermediate sub-assembly configuration choices during assembly that determine which one of the final assembly configurations is more likely to result.

#### 1.1. Assembly Configurational Volume

#### 1.2. Kinetics, Topology and Geometric Complexity

#### 1.3. Recursive Decomposition, Assembly Trees and Combinatorial Entropy

#### 1.4. Symmetry in Chemistry

#### 1.5. EASAL: Efficient Atlasing and Search of Assembly Landscapes

#### 1.5.1. Geometrization

#### 1.5.2. Stratification

#### 1.6. Organization and Contribution

## 2. Framework for Symmetry in an Assembly

**Theorem 1.**

#### 2.1. A Bunch and Its Symmetries

**bunch**is a tuple $(P;\mathcal{C},r,\delta )$ where $P=({p}_{1},{p}_{2},\dots ,{p}_{n})$ is an ordered set of points in ${\mathbb{R}}^{3}$, and $\mathcal{C},r,\delta $ are functions defining colored spheres centered at the points in P. Specifically, $\mathcal{C}:P\to C$ where C is a finite set of “colors”, and $r,\delta :P\to {\mathbb{R}}^{+}$, such that the spheres are non-intersecting, i.e., $\parallel {p}_{i}-{p}_{j}{\parallel}_{2}\ge r\left({p}_{i}\right)+r\left({p}_{j}\right)$ for any $i\ne j$. The map δ is interpreted as the width of the annulus specified by the potential energy well and is used in the definition of an active constraint graph of an assembly configuration later in this section. For a bunch B, $P\left(B\right)$ is used to denote the point set B; similarly, we have $\mathcal{C}\left(B\right),r\left(B\right)$ and $\delta \left(B\right)$.

**isomorphic**if there is an element ϕ of $SE\left(3\right)$ and a permutation $\pi \in {S}_{n}$, such that $\varphi \left({p}_{i}\right)={p}_{\pi \left(i\right)}^{\prime}$ for all i, where $n=\left|P\right|$, and ϕ preserves the color, radius and annulus of points. In this case, with a slight abuse of notation, we write ${B}^{\prime}\in \varphi \left(B\right)$, where $\varphi \left(B\right)$ denotes the set of bunches that are isomorphic to B under ϕ and some permutation in ${S}_{n}$. See Figure 1 for an example.

**strictly isomorphic**, if there is a permutation $\pi \in {S}_{n}$ such that B and ${B}^{\prime}$ are isomorphic under π and the identity element in $SE\left(3\right)$. The

**weak automorphism group**of B, denoted $\mathit{Waut}\left(B\right)$, is the group of all permutations $\pi \in {S}_{n}$ that take B to a strictly isomorphic ${B}^{\prime}$.

**order-preserving isomorphic**or

**congruent**, if there is a $\varphi \in SE\left(3\right)$, such that B and ${B}^{\prime}$ are isomorphic under ϕ and the identity permutation. In this case, with a slight abuse of notation, we write ${B}^{\prime}=\varphi \left(B\right)$.

**Observation 2.**

#### 2.2. An Assembly Configuration Space and Its Symmetries

**assembly configuration**is an ordered set $\mathcal{B}=({B}_{1},{B}_{2}\dots {B}_{k})$, where ${B}_{i}=({P}_{i};{\mathcal{C}}_{i},{r}_{i},{\delta}_{i})$ is a bunch for all i, such that for all $i,j$ and all $x\in {P}_{i},\phantom{\rule{0.277778em}{0ex}}y\in {P}_{j},\phantom{\rule{0.277778em}{0ex}}x\ne y$, we have:

**assembly configuration space**. The assembly configuration space containing the assembly configuration $\mathcal{B}$ is denoted $\mathcal{A}\left(\mathcal{B}\right)$ or simply $\mathcal{A}$ when the context is clear.

**isomorphic**if there is an element ϕ of $SE\left(3\right)$ (isomorphism between bunches) and a permutation $\sigma \in {S}_{k}$, such that for all i, ${B}_{\sigma \left(i\right)}^{\prime}$ is isomorphic to ${B}_{i}$ under ϕ and a permutation ${\pi}_{i}\in {S}_{{n}_{i}}$, where ${n}_{i}=\left|{P}_{i}\right|$.

**strictly isomorphic**, if there is a permutation $\sigma \in {S}_{k}$, such that for all i, ${B}_{\sigma \left(i\right)}^{\prime}$ is isomorphic to ${B}_{i}$ under the identity element in $SE\left(3\right)$ and a permutation ${\pi}_{i}\in {S}_{{n}_{i}}$, where ${n}_{i}=\left|{P}_{i}\right|$. Thus, a strict isomorphism is a tuple of permutations $(\sigma ,{\pi}_{1},\dots ,{\pi}_{k})$, where $\sigma \in {S}_{k}$ and ${\pi}_{i}\in {S}_{{n}_{i}}$. The

**weak automorphism group**of $\mathcal{B}$, denoted $\mathit{Waut}\left(\mathcal{B}\right)$, is the group of all such tuples $(\sigma ,{\pi}_{1},\dots ,{\pi}_{k})$ that take $\mathcal{B}$ to a strictly isomorphic ${\mathcal{B}}^{\prime}$, with the group operation $(\sigma ,{\pi}_{1},\dots ,{\pi}_{k})({\sigma}^{\prime},{\pi}_{1}^{\prime},\dots ,{\pi}_{k}^{\prime})=(\sigma {\sigma}^{\prime},{\pi}_{1}{\pi}_{1}^{\prime},\dots ,{\pi}_{k}{\pi}_{k}^{\prime})$.

**weak automorphism group**of an assembly configuration space $\mathcal{A}$, denoted $\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}$, to be the weak automorphism group of any assembly configuration $\mathcal{B}$ in $\mathcal{A}$.

**congruent**if there is an isomorphism $\varphi \in SE\left(3\right)$ that preserves both the order of the bunches and the order of points within each bunch, i.e., for all i, ${B}_{i}^{\prime}$ is congruent to ${B}_{i}$ under ϕ. Two assembly configurations $\mathcal{B}$ and ${\mathcal{B}}^{\prime}$ are

**strictly congruent**if they are both congruent and strictly isomorphic. In general, we think of two strict congruent assembly configurations as the same. The

**strict congruence group**of an assembly configuration $\mathcal{B}$ is the stabilizer of the set strictly congruent assembly configurations of $\mathcal{B}$ under $\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}$. It is the stabilizer subgroup ${\mathit{stab}}_{\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}}\mathcal{B}$ of the assembly configuration $\mathcal{B}$ under $\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}$.

**order-preserving isomorphic**if there is an isomorphism $\varphi \in SE\left(3\right)$ that preserves the order of the bunches, i.e., for all i, ${B}_{i}^{\prime}$ is congruent to $\varphi \left({B}_{i}\right)$. Two assembly configurations $\mathcal{B}$ and ${\mathcal{B}}^{\prime}$ are

**strictly order preserving isomorphic**if they are both order-preserving isomorphic and strictly isomorphic. The

**strict order-preserving isomorphism group**of an assembly configuration $\mathcal{B}$ is the stabilizer of the set of strictly order-preserving isomorphic configurations of $\mathcal{B}$ under $\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}$.

**permuted congruent**if there is an isomorphism that preserves the order of points within each bunch, i.e., there is an element ϕ of $SE\left(3\right)$ and a permutation $\sigma \in {S}_{k}$, such that for all i, ${B}_{\sigma \left(i\right)}^{\prime}$ is congruent to ${B}_{i}$ under ϕ. Two assembly configurations $\mathcal{B}$ and ${\mathcal{B}}^{\prime}$ are

**strictly permuted congruent**if they are both permuted congruent and strictly isomorphic. The

**strict permuted congruence group**of an assembly configuration $\mathcal{B}$ is the stabilizer of the set of permuted congruent configurations of $\mathcal{B}$ under $\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}$.

**Figure 2.**The assembly configuration ${\mathcal{B}}_{1}$ consists of three isomorphic bunches. ${\mathcal{B}}_{2}$ is obtained from ${\mathcal{B}}_{1}$ with a strict congruence; ${\mathcal{B}}_{3}$ is obtained from ${\mathcal{B}}_{1}$ with a strict permuted congruence; and ${\mathcal{B}}_{4}$ is obtained from ${\mathcal{B}}_{1}$ with a strict isomorphism that is neither a strict congruence, nor a strict permuted congruence, nor a strict order preserving isomorphism.

**Figure 3.**Four assembly configurations obtained by applying $\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}$ on the assembly configuration ${\mathcal{B}}_{1}$. ${\mathcal{B}}_{2}$ is obtained from ${\mathcal{B}}_{1}$ with a congruence, while ${\mathcal{B}}_{3}$ is obtained from ${\mathcal{B}}_{1}$ with a strict order-preserving isomorphism.

**Observation 3.**

- $\mathcal{B}$ and ${\mathcal{B}}^{\prime}$ are strictly congruent if and only if they are congruent, and
- (*)
- $\mathcal{B}$ and ${\mathcal{B}}^{\prime}$ have the same unordered partition of the unordered point set into bunches, i.e., $\{\tilde{{P}_{1}},\dots ,\tilde{{P}_{k}}\}=\{\tilde{{P}_{1}^{\prime}},\dots ,\tilde{{P}_{k}^{\prime}}\}$, where $\tilde{{P}_{i}}$ is the unordered point set of the bunch ${B}_{i}$, and each point has the same color, radius and annulus in $\mathcal{B}$ and ${\mathcal{B}}^{\prime}$.

- $\mathcal{B}$ and ${\mathcal{B}}^{\prime}$ are strictly order-preserving isomorphic if and only if they are order preserving isomorphic and satisfy the condition (*).
- $\mathcal{B}$ and ${\mathcal{B}}^{\prime}$ are strictly permuted congruent if and only if they are permuted congruent and satisfy the condition (*).

#### 2.3. Symmetries in an Active Constraint Graph and an Active Constraint Region

**active constraint graph**$G\left(\mathcal{B}\right)$ of an assembly configuration $\mathcal{B}=({B}_{1},\dots ,{B}_{k})$ is a graph $(V,E)$, where the vertex set V has one vertex for each point $p\in {P}_{1}\cup \dots \cup {P}_{k}$, labeled by a tuple $(i,l)$, representing that the point p appears as the i-th point ${p}_{i}$ in the l-th bunch ${B}_{l}$ of $\mathcal{B}$, and a vertex pair $\{x,y\}\in E$ if x and y lie in distinct bunches of $\mathcal{B}$ and:

**isomorphic**if there is a $\psi =(\sigma ,{\pi}_{1},\dots ,{\pi}_{k})\in \mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}$, such that $\{x,y\}\in E\left({G}_{1}\right)\u27fa\{\psi \left(x\right),\psi \left(y\right)\}\in E\left({G}_{2}\right)$. In this case, we say ${G}_{1}{\cong}_{\psi}{G}_{2}$ or $\psi \left({G}_{1}\right)={G}_{2}$.

**automorphism group**of an active constraint graph G is the group of elements $\psi \in \mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}$, such that $\psi \left(G\right)=G$, i.e., it is the stabilizer subgroup ${\mathit{stab}}_{\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}}G$.

**Figure 4.**All non-isomorphic active constraint graphs with 12 edges of an assembly system of six bunches that are identical singleton spheres. The label on top is automatically generated by EASAL and specifies the orbit number of the shown active constraint graph.

**Note:**

**Figure 5.**An assembly configuration whose automorphism group is strictly contained in that of the corresponding active constraint graph. Here, the bunches are singleton spheres, and bunches of the same color have the same $\mathcal{C}$, r and δ.

**full graph**${G}^{*}$ of an active constraint graph G is obtained by adding edges to G to make the set of vertices in each bunch into a clique.

**active constraint region**${R}_{G}$ of the assembly configuration space $\mathcal{A}$ contains all assembly configurations $\mathcal{B}$ with the active constraint graph $G\left(\mathcal{B}\right)=G$. The action of elements of $\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}$ on an active constraint region and the stabilizer of an active constraint region in $\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}$ are well-defined by the action of $\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}$ on assembly configurations.

**Theorem 4.**

**Proof.**

**Figure 6.**Any assembly configuration corresponding to the active constraint graph G has its strict congruence group strictly contained in ${\mathit{stab}}_{\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}}G$. Here, the bunches are singleton spheres, and bunches of the same color have the same $\mathcal{C}$, r and δ.

**Remark 1.**

#### 2.4. Symmetries in Stratification, Assembly Path and Pathway

**stratification**$\mathcal{S}\left(\mathcal{A}\right)$ of the assembly configuration space $\mathcal{A}$ is a partition of the space into strata ${\mathcal{X}}_{i}$ of $\mathcal{A}$ that form a filtration $\varnothing \subset {\mathcal{X}}_{0}\subset {\mathcal{X}}_{1}\subset \dots \subset {\mathcal{X}}_{m}=\mathcal{A}$, $m=6(n-1)$. Each ${\mathcal{X}}_{i}$ is a union of active constraint regions ${R}_{G}$, where the corresponding active constraint graph G has $m-i$ independent edges, i.e., $m-i$ inequality constraints are active. Each active constraint graph G is itself part of at least one, and possibly many, hence, l-indexed, nested chains of the form $\varnothing \subset {G}_{0}^{l}\subset {G}_{1}^{l}\subset \dots \subset {G}_{m-i}^{l}=G\subset \dots \subset {G}_{m}^{l}$.

**Figure 7.**A fundamental region of the stratification for the assembly configuration space of the assembly configurations in Figure 4 of six bunches, with each bunch being a singleton sphere and all bunches identical. Therefore, $\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}$ is the complete symmetric group of the permutations of six elements, ${S}_{6}$. Each node shown is an orbit representative of an active constraint region corresponding to an active constraint graph. The grey part is those active constraint graphs (orbit representatives) whose corresponding constraint regions are empty. The example active constraint graph representatives on the right have arrows pointing to their regions in the stratification. The labels in the circles are unimportant: they are automatically generated and specify an orbit of an active constraint graph (example shown on the right).

**assembly path**from ${G}_{1}$ to ${G}_{m}$ in the stratification is a sequence ${G}_{1}\u228a{G}_{2}\u228a{G}_{3}\u228a\dots \u228a{G}_{m}$ where ${G}_{i+1}$ is a child of ${G}_{i}$ for all $1\le i\le m$. A

**coarse assembly path**from ${G}_{1}$ to ${G}_{m}$ in the stratification is a sequence ${G}_{1}\u228a{G}_{2}\u228a{G}_{3}\u228a\dots \u228a{G}_{m}$ where ${G}_{i+1}^{*}$ has exactly one new rigid component S not in ${G}_{i}^{*}$, with S containing a set of two or more rigid components ${S}_{1}\dots {S}_{m}$ of ${G}_{i}$. In addition, for all proper subsets $Q\u228a\left\{{S}_{1}\dots {S}_{m}\right\}$ with $\left|Q\right|\ge 2$, the subgraphs of ${G}_{i+1}^{*}$ induced by Q are not rigid (The rigid components of a graph are the maximal rigid subgraphs. Two rigid components cannot intersect on more than two vertices. We refer the reader to combinatorial rigidity concepts in [61]).

**assembly forest**corresponding to a coarse assembly path from ${G}_{1}$ to ${G}_{m}$ is the unique forest where the leaves are the maximal rigid components of ${G}_{1}^{*}$. The internal nodes are the new rigid components S occurring in some ${G}_{i+1}^{*}$ in the path. The children of S are the set of rigid components ${S}_{1}\dots {S}_{m}$ contained in S that occur in ${G}_{i}^{*}$. The roots of the forest are the rigid components of ${G}_{m}^{*}$. An

**assembly tree**is an assembly forest with only one root. See Section 3 for examples of assembly trees [46,49,62].

**full (coarse) assembly path**is an (coarse) assembly path from ${G}_{1}$ to ${G}_{m}$, where ${G}_{1}$ is the empty active constraint graph, and ${G}_{m}^{*}$ is a rigid active constraint graph. A

**(coarse) assembly path from primitives**has the first property of the full assembly path, i.e., ${G}_{1}$ is the empty active constraint graph, but not the last property, i.e., ${G}_{m}$ can be any active constraint graph. The

**full assembly tree**and assembly tree from primitives are also defined in this way.

**path**between full active constraint graphs G and H where $G\u2288H$ and $H\u2288G$ is a sequence $G={G}_{i},{G}_{i+1},{G}_{i+2},\dots ,{G}_{i+m}=H$, where any pair ${G}_{i+k}$ and ${G}_{i+k+1}$ is on some assembly path, and ${G}_{i+k}\u228a{G}_{i+k+1}$ if k is even, ${G}_{i+k}\u228b{G}_{i+k+1}$ if k is odd.

**fundamental domain**of the stratification $\mathcal{S}\left(\mathcal{A}\right)$ is the minimal sub-stratification $\tilde{\mathcal{S}}\left(\mathcal{A}\right)$, such that ${\bigcup}_{\pi \in \mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}}\pi \left(\tilde{\mathcal{S}}\left(\mathcal{A}\right)\right)=\mathcal{S}\left(\mathcal{A}\right)$, where π acts on $\tilde{\mathcal{S}}\left(\mathcal{A}\right)$ via its action on the active constraint regions (resp. active constraint graphs) of $\tilde{\mathcal{S}}\left(\mathcal{A}\right)$. In other words the active constraint regions (resp. active constraint graphs) in $\tilde{\mathcal{S}}\left(\mathcal{A}\right)$ are orbit representatives of active constraint regions (resp. active constraint graphs) under $\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}$.

**assembly pathway**is an orbit of an assembly tree under $\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}$. The definition extends to full and coarse assembly trees.

#### 2.5. Example Illustrating the above Symmetries

**Example 1.**

**Figure 8.**The neighbors of one active constraint graph in the Hasse diagram of the stratification for the assembly system in Figure 4.

## 3. Enumerating Simple Assembly Pathways

**simplified assembly tree**is a rooted tree for which each internal vertex has at least two children and whose leaves are bijectively labeled with elements of a set X. There is an induced labeling on all of the vertices of a simplified assembly tree by labeling a vertex v by the set of labels on the leaves that are descendents of v. We identify each vertex of a simplified assembly tree with its label. Two simplified assembly trees are considered identical if there is a root-preserving, adjacency-preserving and label-preserving bijection between their vertex sets. The 26 simplified assembly trees with four leaves, labeled in the set $X=\{1,2,3,4\}$, are shown in Figure 9.

**simplified assembly pathway for**$(G,X)$.

**Example 2.**

**Theorem 5.**

**Theorem 6.**

**Example 3.**

**Theorem 7.**

**Remark 2.**

**Example 4.**

**Example 5.**

- the identity,
- 15 rotations of order 2 about axes that pass through the midpoints of pairs of diametrically opposite edges of P,
- 20 rotations of order 3 about axes that pass through the centers of diametrically opposite triangular faces and
- 24 rotations of order 5 about axes that pass through diametrically opposite vertices.

- 15 subgroups of order 2, each generated by one of the rotations of order 2,
- 10 subgroups of order 3, each generated by one of the rotations of order 3,
- 5 subgroups of order 4, each generated by rotations of order 2 about perpendicular axes,
- 6 subgroups of order 5, each generated by one of the rotations of order 5,
- 10 subgroups of order 6, each generated by a rotation of order 3 about an axis L and a rotation of order 2 that reverses L,
- 6 subgroups of order 10, each generated by a rotation of order 5 about an axis L and a rotation of order 2 that reverses L,
- 5 subgroups of order 12, each the symmetry group of a regular tetrahedron inscribed in P.

## 4. Open Questions

#### 4.1. Enumeration Problems in (Non-Simplified) Assembly Framework

- (1)
- How does one compute the size of orbits/stabilizers and the number of orbits under $\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}$ for assembly configurations, active constraint graphs, active constraint regions, (coarse) assembly paths and assembly trees/forests?
- (2)
- How does one compute the number of coarse assembly paths that correspond to a particular assembly tree/forest?
- (3)
- Given two active constraint graphs G and H, where G and H are incomparable, i.e., $G\u2288H$ and $H\u2288G$, how does one compute the number of paths between them?
- (4)
- Given two active constraint graphs ${G}_{1}$ and ${G}_{m}$, where ${G}_{1}\u228a{G}_{m}$, how does one compute the number of (coarse) assembly paths from ${G}_{1}$ to ${G}_{m}$?
- (5)
- What are the orbits of the (coarse) assembly paths in (4) under the action of ${\mathit{stab}}_{\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}}\left({G}_{m}\right)$?
- (6)
- What are the orbits of the (coarse) assembly paths in (4) under the action of the group H, where $H=\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}$ if ${\mathit{stab}}_{\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}}\left({G}_{1}\right)=\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}$ (i.e., ${G}_{1}$ is the empty active constraint graph), or $H={\mathit{stab}}_{\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}}\left({G}_{1}\right)\cap {\mathit{stab}}_{\mathit{Waut}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathcal{A}}}\left({G}_{m}\right)$ otherwise?

#### 4.2. Symmetries within an Active Constraint Region via Cayley Configurations

**Theorem 8.**

**Theorem 9.**

#### 4.2.1. Fundamental Regions of Active Constraint Regions

#### 4.3. g-Unfixable Unlabeled Trees

#### 4.4. Depth of an Assembly Pathway

#### 4.5. Other Questions

- Given two symmetry-invariant properties, how does one compute the ratio of the number of pathways that satisfy both of these properties to the number of symmetry classes that satisfy only one of these properties?
- What can we say about larger (icosahedrally) symmetric polyhedral graphs (larger T numbers of viral capsids, for example), fullerenes and fulleroids and polyhedra with different symmetry groups? In such cases, the computations of Section 3 can also be phrased as algorithmic questions, where the asymptotic complexity of the algorithm is expressed in terms of the number of facets of the polyhedron (or the T number).
- To fully extend the techniques in Section 3 to the framework of Section 2, each sub-assembly must be a rigid subgraph of the graph at the root. Some assembly trees fail to satisfy the rigidity condition and can never occur (probability zero). Such assembly trees are geometrically invalid. In addition, a valid assembly tree can be assigned a non-zero probability according to how difficult it is to find a solution to the constraints on each sub-assembly. Computing this probability, called the geometric stability factor, is necessary to make the required predictions.Dropping the rigidity requirement, but maintaining the subgraph (connectivity) requirement, in [79], two of the authors study the number of assembly trees of graphs on labeled vertices. In that model, each graph has a trivial automorphism group, but the enumeration of assembly trees still leads to the use of a recent and very powerful technique from the theory of D-finite power series in several variables.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Sitharam, M.; Vince, A.; Wang, M.; Bóna, M.
Symmetry in Sphere-Based Assembly Configuration Spaces. *Symmetry* **2016**, *8*, 5.
https://doi.org/10.3390/sym8010005

**AMA Style**

Sitharam M, Vince A, Wang M, Bóna M.
Symmetry in Sphere-Based Assembly Configuration Spaces. *Symmetry*. 2016; 8(1):5.
https://doi.org/10.3390/sym8010005

**Chicago/Turabian Style**

Sitharam, Meera, Andrew Vince, Menghan Wang, and Miklós Bóna.
2016. "Symmetry in Sphere-Based Assembly Configuration Spaces" *Symmetry* 8, no. 1: 5.
https://doi.org/10.3390/sym8010005