The determinism and boundedness of selfassembling structures
Abstract
Selfassembly processes are widespread in nature, and lie at the heart of many biological and physical phenomena. The characteristics of selfassembly building blocks determine the structures that they form. Among the most important of these properties are whether the selfassembly is deterministic or nondeterministic, and whether it is bound or unbound. The former tells us whether the same set of building blocks always generates the same structure, and the latter whether it grows indefinitely. These properties are highly relevant in the context of protein structures, as the difference between deterministic protein selfassembly and nondeterministic protein aggregation is central to a number of diseases. Here we introduce a graphbased approach that can determine, with a few restrictions, whether a set of selfassembly building blocks is deterministic or nondeterministic, and whether it is bound or unbound. We apply this methodology to a previously studied lattice selfassembly model and discuss generalisations to a wide range of other selfassembling systems.
I Introduction
Selfassembly is an ubiquitous phenomenon in nature, producing complex structures in biology, chemistry and physics. Examples include DNA winfree1998design (); mao2000logical (); chworos2004building (); goodman2005rapid (); rothemund2006folding (); fujibayashi2007toward (); ke2012three (), protein quaternary structure Levy:2006ez (); villar2009self (), protein aggregation Cohen:2013fd (), viruses zlotnick1994build (), micelles israelachvili1994self (), and thin films krausch2002nanostructured (). Perhaps the two most fundamental properties of selfassembling systems are (a) whether the building blocks always form the same structure or not, and (b) whether the selfassembling structure grows indefinitely or remains bounded. The former property is sometimes referred to as the determinism and the latter as boundedness Evo (); ARXIV2015 (). Protein complexes are a prominent example of deterministic, bound selfassembly. Misfolding or erroneous binding of mutated versions of such proteins can cause the selfassembly of a protein complex to become nondeterministic, unbound protein aggregation, which in turn is the hallmark of a number of severe diseases, such as sicklecell anaemia, and Alzheimer’s Cohen:2013fd (); Bunn:1997de (). In this paper we introduce a framework for establishing whether a given set of selfassembly building blocks is deterministic or nondeterministic, and whether it is bound or unbound. We apply this approach to a previously studied lattice selfassembly model introduced in Modularity () and studied further in Evo (); green (); greengen (); ARXIV2015 (); TES2016 (). In this model square tiles have attractive interfaces of different types. The interfaces can be symmetric or asymmetric, and the interface type is denoted by an integer (sometimes termed the ‘colour’ of the interface). The interactions between colours are specified by a binary interaction matrix, the assembly process is stochastic, often starting from a single seed tile, and the final structures assembled in this way represent sets of connected lattice sites that are sometimes referred to as Polyominoes. Such tile selfassembly or Polyomino models have been useful for the study of genotypephenotype maps, where the specification of the building blocks can be viewed as a genotype and the resulting structure as the respective phenotype Evo (); green (); greengen (). This approach can be combined with genetic algorithms to model evolutionary processes Evo (). Despite being a very abstract model, real biological selfassembly phenomena can be directly and meaningfully mapped to this Polyomino model. For example, the sicklecell mutation of haemoglobin that leads to unbound protein aggregation can be modelled using Polyominoes green (). Furthermore, experimental implementations of the Polyomino model have already been realised using DNA tiles TES2016 ().
The graphbased approach introduced in this paper aims to identify whether a given set of selfassembly building blocks will produce a final structure deterministically or nondeterministically, and whether that structure will be bound or unbound. The algorithm can be generalised to building blocks of any regular geometry and any number of binary attractive interactions. It therefore has potential applications in the study of protein complexes and protein aggregation, as well as bioengineering and nanotechnology.
Ii Lattice selfassembly
Following the model introduced in Modularity () a lattice selfassembly tile set consists of one or more tiles types. Each tile has interactive interfaces on some or all of its four faces, and different tile types exhibit different configurations of these interfaces. The interfaces, sometimes referred to as ‘colours’ Evo (); Modularity () can be represented as integers, and interactions between them can be described by a binary interaction matrix. The interactions can be symmetric, when an interface binds to another of the same type, (e.g. ) or asymmetric, when it binds to an interface of a different type (e.g. ), while 0 denotes a neutral interface. The interactions are infinite in strength, meaning that if two tiles come into contact, they will bind to each other irreversibly if the two faces in contact have interactions that are compatible according to the interaction matrix. The assembly algorithm is fully stochastic. After the assembly is seeded with one random tile from the tile set, the assembly process undergoes the following steps: (a) a random face on the seed is chosen (b) a new tile is selected randomly from the tile set, (c) attachment to the selected face on the seed tile is attempted with a random orientation of the newly selected tile, and (d) if the interfaces on the faces in contact are compatible, the new tile is attached. Steps (ad) are then repeated on a random face on the structure that is being formed until there are only neutral (interaction type 0) faces on the perimeter of the structure, in which case it cannot grow any further. Note that cooperative binding is not possible in our algorithm. Unless stated otherwise, we will assume asymmetric interactions for the remainder of this paper, such that each odd interface type interacts with its conjugate type, for example . See Figure 1 for an illustration of a simple selfassembly tile set.
Iii Definitions of Structure and Determinism
Definition of a tile structure: A structure is defined in terms of the relative positions and orientations of the building block instances, and the building block types. The definition of a structure is therefore independent of its absolute position or orientation as a whole on the lattice. Note that the seed tile is chosen randomly, which means that most structures will at least vary in terms of these degrees of freedom, even if they are otherwise deterministic.
Deterministic assembly tile set: An assembly tile set that always produces the same structure (in the limit for unbound assemblies).
There are several variations of this definition of determinism that can be adopted within this lattice model. These are listed below. The first two are less strict than the above definition, and the last one is more restrictive.
Type and orientationindependent determinism: Ignoring the tile types and orientations in the structure, the only criterion is that the same overall shape of the structure is produced by the assembly algorithm.
Orientationindependent determinism: Ignoring the tile orientations, the tile types and overall shape of the structure have to be the same in every assembly.
Absolute determinism: The structure is completely determined by the tile set, even with respect to its absolute position and orientation.
This paper focuses on the simplest definition of determinism, which is the first one given above, requiring tile types, orientations, and relative positions to be fixed, but leaving the absolute position and orientation unspecified, as the random seed would otherwise render most tile sets nondeterministic.
Iv Assembly graph
The assembly graph of a tile set is a graphical representation of the interaction network between tiles with interactive faces. The connections in this network specify pairs of tile faces whose interfaces interact. Any two faces can at most have one connection between them. Figure 1 (left) shows an example of an assembly graph.
A branching point in an assembly graph is defined as a tile face that connects to more than one other tile face. Branching points are the main source of nondeterminism because the stochastic choice of tiles during the assembly process means that one could choose one of several different tiles to successfully bind to a given face. However, branching points can also be deterministic. For instance, a tile with a single interactive face (SIF) can bind deterministically to several other tiles, e.g. has a branching point, but is a deterministic tile set. Branching points will be studied in greater detail below. An example of a structure with a branching point is presented in Figure 2.
V Graph jointness
We do not allow a graph to be disjoint, as this would lead to seed dependence TES2016 (), which means that a different final structure would emerge for different seed tiles.
Vi Treelike assembly graphs without branching points are ruledeterministic
In a treelike assembly graph without branching points, each tile face interacts only with a single other active face in the tile set, as shown in the example in Figure 1.
We define an assembly graph walk as a walk along the interaction graph, which consists of two alternating steps: Starting on a connected tile face the first step is to move along a connection in the graph to another tile face. The second step is to move to a different connected face on the same tile, before proceeding again with the first step. This walk mirrors the assembly process, as the first step represents the binding of two tiles, and the second step reflects the fact that after binding a face is unavailable to bind another tile.
Note that the assembly graph formalism does not allow the detection of steric effects, whereby a structure grows into itself. It therefore only allows us to determine whether a tile set is ’ruledeterministic’, meaning deterministic according to its interaction rules. A ruledeterministic tile set can still be nondeterministic due to steric effects. This limitation will be discussed in more detail below.
Tile sets with treelike assembly graphs that have no branching points are deterministic up to steric effects, because each active face can only bind to one other tile face, and each tile has a list of neighbours uniquely defined, up to steric effects. It should also be noted that, if the assembly graph is treelike without branching points, there will always exist at least two tiles in the tile set with only one single interactive face (SIF tiles). This observation can be seen as a parallel to the theorem that every nonedgeless finite tree has two leaf nodes math3 (). These properties of treelike assembly graphs will be used in the analysis of more complex assembly graphs later on.
Vii Alternative treelike assembly procedures
There can be several assembly avenues, giving rise to identical overall lattice shape (think of the different ways a cross can be produced, see Figure 3). These assemblies will not code for the same structure as defined above (in terms of tile types, positions, and orientations) but will result in type and orientationindependent determinism. See figure 3 for clarifications.
In general a branching point tile with a Single Interactive Face (SIF tile), whose colour on the interactive interface does not appear anywhere else in the tile set possesses an alternative treelike representation. To realise such alternative representation, introduce as many new SIF tiles in the tile set as the number of faces that colour binds to, and connect them individually to each of the faces that the SIF tile branching point originally connects to.
We also note that if two interacting colours are both present on more than one interactive face each, then assembly is nondeterministic.
These alternative representations are important in the context of the next section which describes a way to recognise determinism from inspection of the selfassembly tile set.
vii.1 SIF tiles elimination procedure
Consider a graph with a SIF tile connecting to several faces in the assembly. As shown in top image of figure 3, such a SIF branching point tile can be replaced by treelike assembly avenues. Based on this we introduce the following procedure for eliminating SIF tiles in order to study the deterministic nature of a tile set:
SIF tiles elimination procedure: Transform SIF branching point tiles (whose colour appear only once in the whole tile set) into treelike assembly representation as outlined in the section above. Then, proceed with eliminating SIF tiles from the assembly tile set, turning into 0s the colours of the tiles the SIF tiles bind to.
If the graph is treelike and therefore deterministic, through the elimination procedure described above, a deterministic selfassembly tile set with no loops will always be reduced to two connected SIF tiles.
Viii Loop graphs
Loop graph or subgraph: A loop is a set of tiles that give rise to a cyclic assembly graph walk, which means that, given a subset of tiles from an assembly tile set of tiles (), if starting from any of the tiles in the subset there exists one assembly graph walk that leads in steps back to the first tile in the assembly subset with each tile being traversed once throughout the walk, then this set of tiles forms a loop graph () or loop subgraph().
viii.1 Loop rank observations
The rank of a loop defines the number of times each tile in the loop is used during the assembly of the structure. In a loop, after all tiles in the loop have been used exactly once during assembly, reattachment of the tile can happen with an orientation shift by an angle , or . Therefore, in two dimensions, there can be four types of loops:
Rank 1: in loops of rank 1, each tile is used exactly once. This means that seeding the assembly loop randomly from any tile in the selfassembly tile set, the last tile in the loop reattaches to the seed with matching tile faces.
Rank : the seed is to be reattached with no orientational shift, then the structure will keep growing forever, such as the single selfassembly tile .
Rank 2: if the seed tile is reattached with , each tile will appear altogether twice before the structure reattaches to the initial seed tile location, implying that the loop is of rank 2.
Rank 4: Similar to the argument for rank 2 loops, if , the loop is of rank 4.
Loop rank examples can be found in figure 5.
viii.2 Establishing boundedness
If there are at least 2 loops of rank higher than 1, growth becomes unbound.
As an example, consider the two loops shown in figure 6: loop and loop have rank 4. Each time an assembly of loop is performed, tile will be assembled four times in the structure. Loop in turn, will create copies of tile another 4 times, triggering assembly of loop and so on. Hence the two loops keep regenerating each other.
viii.3 Eliminating loops of rank 1
Loops of rank 1 possess alternative treelike assembly avenues, in a similar fashion to SIF branching point tiles as described above. For this reason, we can directly turn loops of rank 1 into their alternative treelike description by eliminating one of the edges in the loop before the SIF elimination procedure. See figure 7 for an example.
If after SIF elimination, loops of rank higher than 1 are present along with irreducible branching points, assembly is nondeterministic. An example of this can be found in figure 8. Branching points would have to be traversed stochastically during assembly and neighbours to a tile face would not be uniquely defined.
Ix Symmetry
So far, we have disregarded symmetric tiles because they produce branching points of a different nature from the branching points analysed above and more care has to be taken. In fact, we can get rid of branching points resulting from geometrical symmetry in the tiles via the introduction of alternative assembly descriptions.
Consider for instance the tile set which produces unbound deterministic growth. We can introduce alternative deterministic assembly pathways getting rid of symmetry explicitly, yet preserving the same growth behaviour, as shown in the example in figure 9. Full details on the symmetrisation procedure are included in the appendix.
X Steric Effects
In the preceding sections we have addressed the question whether the assembly interactions alone make a tile set deterministic or nondeterministic, and bound or unbound. However, steric effects can also affect the growth of structures and produce sterically bound structures that are unbound according to the interaction rules or sterically nondeterministic structures that are deterministic according to the rules. In order to recognise steric effects, we can grow the structure a single time: if any coloured faces are mismatched during growth, and if the complementary colour of at least one of these appears elsewhere in the assembly set, then some assembly avenues are sterically prevented. See Figure 10. Therefore, once ruledeterminism has been detected, growing the structure once allows for the detection of steric effects.
Xi Final Procedure
The flowchart in figure 11 illustrates all of the steps to be implemented when determining the (non)deterministic (un)bound nature of a selfassembly tile set.
Xii Discussion
xii.1 Other dimensions and geometries
The algorithms presented in this paper for recognising tile selfassembly determinism can be directly extended to other regular geometries. In extending the determinism rules, the loop rank theorems need to be adjusted, taking into account the appropriate shifts introduced during structure assembly. The plane can regularly be tiled using triangles, squares and hexagons among the regular polygons 1977 ().
The same theorems will apply in the reduction procedure, and one will simply have to consider that for triangles there can only be loops of finite rank 1, 2 and 6 while for hexagons, there can be loops of finite rank 1,2,3,4 and 6. Some loop examples for regular hexagons are presented in figure 12.
Similarly, an extension of the algorithms can be performed in 3 dimensions. For instance for cubes, the finite loop ranks can still be only 1,2 and 4, as for squares because the possible orientational shifts can only be with or . Therefore, we recognise that the deterministic rules derived in this paper can be extended to higher dimensions and other regular geometries.
xii.2 Nondeterministic genotypephenotype maps
In ARXIV2015 (), we studied in detail nondeterministic selfassembly of single lattice tiles and mixtures of two tiles at varying relative concentrations with colours 0, 1 and 2, where 1 interacts with 2, and 0 is a neutral colour. We were able to classify tile sets in terms of their behaviour upon variation of tile density, recognising critical transitions from bound to unbound growth behaviours and non critical density transitions. In total, 106 topologically distinct tile sets were identified. Tile graph methods now allow us to determine with no need of direct assembly the (un)bound and (non)deterministic tile set behaviour, completing the picture. In figure 13 we present the bound deterministic and unbound deterministic tile sets with two tiles and 3 colours (0, 1 and 2) under asymmetric interactions. The number of deterministic tile sets is much smaller than the nondeterministic ones, 10 versus 106.
Furthermore, we also provide a table with all 106 tile sets and their corresponding tile graphs in the supplementary material for symmetric and asymmetric interactions.
xii.3 Genotypephenotype maps with 2 tiles and 8 colours
The lattice selfassembly model described here has been used as an abstract model of the genotypephenotype map of protein quaternary structure green (). In these models the interactions of the building block set constitute the genotype and the assembled structure represents the phenotype. This work focused on deterministic assembly, while nondeterministic building block sets were regarded as a single unfavourable phenotype, and largely ignored. The assembly graph framework allows us to define the assembly graphs as the phenotypes, and thus explore the genotypephenotype map of nondeterministic tile sets. As nondeterministic selfassembly in the form of protein aggregation is a hallmark of numerous diseases, the study of such nondeterministic genotypephenotype maps can help us to better understand these phenomena.
Here we consider the assembly graph genotypephenotype (AGGP) for two tiles and eight colours. Figure 14 shows the frequency of a phenotype versus the rank of its frequency within the map, which reveals a similar heavytailed distribution to those described in green (). In contrast the Polyomino GP map of deterministic lattice selfassembly, we are now accounting for nondeterministic structures in our AGGP map as well.
The total number of topologically distinct assembly graphs with two tiles and eight colours is 1136. Of these, twelve yield bound deterministic structures, which make up 5876685 of all possible x degenerate genotypes, or 36.7% of all genotype space. In Evo (), the same genotype space has been studied, focusing on the GenotypePhenotype map between the selfassembly tile set and the final assembled structure. In this study, 38.9% of all genotypes were classified as bound and deterministic, through repeated direct assembly of the phenotype from the same seed. The number of bound deterministic genotypes is lower for the AGGP map because steric determinism is considered as a form of nondeterministic behaviour, whereas the GP map in Evo () classifies selfassembly tile sets as deterministic irrespective of mismatches between tile face colours, provided that multiple assemblies of the same selfassembly tile set yield the same final structure. The discrepancy in number of bound deterministic phenotypes is also caused by the seeding choice, which is stochastic for the selfassembly tile sets studied in the AGGP map, and specified for the models explored in Evo ().
In addition to general genotype phenotype maps, the assembly graph framework can also be used to illustrate the function of the sickle cell mutant as modelled using the Polyomino model in green (). In this representation the and chains of wildtype haemoglobin Hb A can be mapped to the  tile set, and the sicklecell mutant Hb S can be represented by the  . The point mutation through the introduction of colour ‘6’ results in unbound deterministic growth giving rise to the sickle cell animia disease. Using the assembly graph framework we can see that tile set  possesses a loop of rank 2 and that the mutation of the second tile, which results in the tile set , leads to the introduction of a second loop into the assembly graph, of rank 4. We can take this further, noting that biological systems that are assembled through loops of rank higher than 1 are prone to this type of mutations to unbound growth via the introduction of a further single loop of rank higher than 1 that is added.
We also suggest that protein misfolding and resulting changes of interface angles in quaternary structures can be recast in these terms, as a loop of rank infinity that is introduced or a loop of rank higher than 1 that is added.
xii.4 Selflimiting cluster growth and limitations
In previous experimental work TES2016 (), we have compared singleseed and multiseed selfassembly, observing that asymmetric interactions can lead to selflimiting growth in multiseed assembly through local steric effects.
The graph methods developed here cannot predict a priori steric effects and therefore can never predict whether an assembly tile set will run into selflimiting growth because of such geometrical constraints.
Xiii conclusion
With the assembly graph framework we have introduced a new approach for the study of selfassembling systems that can help us to determine whether selfassembly outcomes are bound or unbound, and deterministic or nondeterministic.
This also offers a powerful tool for the study of Genotype Phenotype maps, by expanding the exploration of phenotypes to the nondeterministic realm, which offers a very general model of dysfunction and disease in the context of biological selfassembly.
Appendix A Appendix: Symmetry removal
Symmetric tiles in an assembly selfassembly tile set can be substituted with alternative assembly pathways (see Figure 9) eliminating branching points which arise because of tile symmetry and which do not lead to nondeterministic structure growth. Here we present a detailed description for each possible combination of symmetric tiles appearing in the selfassembly tile set.
Symmetric tiles can only be of type , , (where interacts with , with ) and , indicating with any other colour that does not interact with any of the colours.
Observe that having tile in the tile set either leads to seed dependence if no other or colours are present on other interacting tiles the tile set, or it leads to nondeterminism if any or colours appear on other tiles. For this reason, the possibility of having a tile is excluded from this discussion as it should never be taken into account in the symmetrisation procedure.
Similarly, the presence of tiles and leads to seed dependence or nondeterminism if there are any other tiles in the tile set. Nevertheless, the single tile made of alone and tile set  are unbound and deterministic. We exclude the presence of these cases in an assembly tile set from the remainder of our symmetrisation procedure.
Then, defining as any colour which does not interact with or , there exist only 5 possible tile geometries containing a B face: , , , , , if we ignore tiles containing both and colours at the same time as they lead to nondeterminism directly when interacting with any of the , tiles.
Only a choice of either of the or tiles, and any of the , or tiles is deterministic (colour has been chosen without loss of generality as a colour not interacting with which can also be a neutral colour).
Furthermore, tile does not need to be considered because it would lead to redundant cases, whether interacts or not with any of the colours. All cases with colours and swapped have also been ignored because they would lead to redundant combinations.
Our aim is to convert branching points deriving from symmetric interactions into alternative deterministic assembly descriptions, introducing new tiles and colours replicating the behaviour of the original sets.
Therefore, perform the following transformations, which are graphically presented in Figure 16:

1) Convert the pair  into 3 new tiles displaying colours , where colour interacts with colour , with and so on.

2) Similarly,  becomes .

3)  and  remain unchanged because they are deterministic branching points and get reduced directly in the SIF elimination procedure.

4)  becomes .

5) In the presence of two or more symmetric tile cases, Steps need to be combined to remove branching point resulting from symmetric tiles. For instance, in tile set , step 1 and 4 have to be combined, and the symmetrisation procedure results in .
After reducing branching points resulting from tile symmetries leading to deterministic selfassembly, the assembly tile sets can be studied through the procedure described in the main text to assess its bound deterministic nature.
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