Abstract

One of the practical challenges facing the creation of self-assembling systems is being able to exploit a limited set of fixed components and their bonding mechanisms. The method of staging divides the self-assembly process into time intervals, during which components can be added to, or removed from, an environment at each interval. Staging addresses the challenge of using components that lack plasticity by encoding the construction of a target structure in the staging algorithm itself and not exclusively in the design of the components. Previous staging strategies do not consider the interplay between component physical features (morphological information). In this work we use morphological information to stage the self-assembly process, during which components can only be added to their environment at each time interval, to demonstrate our concept. Four experiments are presented, which use heterogeneous, passive, mechanical components that are fabricated using 3D printing. Two orbital shaking environments are used to provide energy to the components and to investigate the role of morphological information with component movement in either two or three spatial dimensions. The benefit of our staging strategy is shown by reducing assembly errors and exploiting bonding mechanisms with rotational properties. As well, a doglike target structure is used to demonstrate in theory how component information used at an earlier time interval can be reused at a later time interval, inspired by the use of a body plan in biological development. We propose that a staged body plan is one method toward scaling self-assembling systems with many interacting components. The experiments and body plan example demonstrate, as proof of concept, that staging enables the self-assembly of more complex morphologies not otherwise possible.

1 Introduction

In the words of Lewis Wolpert [36], “Development is essentially the emergence of organised structures from an initially very simple group of cells.” Construction and self-organization are central to biological development, where the formation of an organism emerges from the interplay between proteins, genes, cells, and the environment [20]. Explicit stages arise from the temporal and positional aspects of the various processes constituting biological development.

In addition to staged biological development, comprehending the principles of self-assembly has been described as one of the important aspects of understanding life [17]. Self-assembly is also considered as an enabling technology for the creation of artificial systems [25]. Constructing systems with natural characteristics (e.g., self-assembly, self-repair, and parallel construction) as a form of emergent engineering requires an understanding of the interplay between programmability/controllability and self-organization [14].

One important challenge when creating artificial self-assembling systems is caused by the use of components that lack the plasticity of biological cells. Using components that cannot differentiate causes self-assembly to be constrained to a limited set of fixed components and their bonding mechanisms [12]. One strategy to address this challenge is to divide the self-assembly process into stages, resulting in staged or hierarchical self-assembly. The method of staging has been formalized [12] in such a way that components can be added to, or removed from, an environment at various time intervals.

The benefits of staging have been demonstrated theoretically using abstract tiles, whereby staging the self-assembly process was based on the temporal aspects of conducting laboratory experiments [12]. In contrast, we use physical components, and propose using morphological information as the dividing basis to staging the self-assembly process [7], inspired by the temporal and positional aspects of staged biological development. Here we demonstrate how physical features in a set of heterogeneous, passive, mechanical components can be used to reduce potential assembly errors, exploit rotational bonding mechanisms, and leverage a limited set of physical features to create target structures with symmetrical/asymmetrical properties. Furthermore, we demonstrate how the concept of a body plan from biological development can be used as one method toward scaling self-assembling systems with many interacting components. Our staging strategy is consistent with the definition of self-assembly [33] as a process involving components that can be controlled through their proper design and their environment, and where components can adjust their relative positions.

Staged self-assembly provides the advantage of encoding the construction of a target structure in the staging algorithm itself and not exclusively in the design of the components. For example, a staging algorithm can be used to reintroduce previously used components and bonding mechanisms at later time intervals, prevent the formation of holes, and create more complex morphologies that may otherwise be impossible due to shape conflicts between components.

The following section presents background material which our staging strategy is built upon. The background material is used to differentiate our work, and includes inspiration from biological development, theoretical foundations of self-assembly, and related physical self-assembling systems. Next, an overview of our approach is provided, including a set of self-assembly rules, a theoretical model, a description of the components and environments used in physical systems, and an analytical comparison of the non-staged versus the staged self-assembly process to justify our approach. Four experiments follow that demonstrate the creation of self-assembled structures, from a set of components that are divided into two time intervals according to their physical features. Components are fabricated using 3D printing [9], and are placed in one of two orbital shaking environments (on a tray surface or in a jar of fluid). These two environments are used to demonstrate the role of morphological information in spatial component movement in two and three dimensions (2D and 3D). Penultimately, the concept of a body plan is used to demonstrate the theoretical self-assembly of a doglike target structure using six time intervals, by reusing morphological information used in previous time intervals. We conclude by summarizing how this work provides proof-of-concept evidence for staging the self-assembly process using morphological information, inspired by biological development.

2 Background

In this section, we first present concepts from biological development that serve as inspiration for our approach to staging the self-assembly process. Next, we outline an abstract model for self-assembly and its staged extension, used to investigate the algorithmic nature of the self-assembly process. Finally, we give examples of physical, staged systems to contrast our staging approach.

2.1 Biological Development

Biological development utilizes explicit stages in its provision of a solution to the construction of multicellular organisms [36]. The explicit stages in biological development, such as invagination, gastrulation, and the formation of a body plan, are often irreversible and cannot be repeated at later stages. Staged development in nature allows for the creation of more complex phenotypes, which otherwise would not be possible [36].

Of the processes involved in biological development, we focus on pattern formation and morphogenesis in the context of the research presented here. Pattern formation is the process by which a spatial and temporal pattern (form or function) of cell activities is organized. For example, pattern formation is involved in initial body planning in embryos, resulting in the creation of a coordinated system based on three axes (anterior-posterior, dorsal-ventral, and left-right with respect to a body). One mechanism for how cells acquire positional identity and value (in relation to an axis and boundary constraints) is using a diffusing morphogen gradient. Morphogenesis is a process resulting in the 3D shape of an organism (e.g., utilizing cell migration in embryo development).

In biological development, staging results from environmental factors of cells triggering the activation and suppression of various gene regulatory networks within cells at different times and locations. Computer models for evolutionary developmental biology have been developed [20], where artificial evolution is used to design a solution and development is used to construct the solution. For example, structures have been evolved in software that model development and cellular differentiation, which were able to self-repair when damaged [21].

2.2 Algorithmic Self-Assembly

A challenge to the creation of self-assembling systems is the use of fixed components in contrast to components that can differentiate and communicate (e.g., cells in biological organisms). DNA nanotechnology and DNA computing are two applications using fixed components. At the nano scale, DNA is considered as one of the most promising materials for the creation of nanotechnology, due to its inherent self-assembly properties [30, 31]. DNA nanotechnology was invented by Nadrian Seeman [25], who realized that 3D lattices could be used to direct molecules, simplifying their crystallographic study. One class of nano-scale components, DNA tiles, were developed to create lattice structures [35]. DNA tiles use interwoven double-stranded DNA to create the square-shaped body of a tile, and single DNA strands extend from the edges of a tile's body [35]. In DNA computing, DNA is manipulated to form shapes (e.g., DNA tiles) that can interact and self-assemble to physically compute functions [24]. Developing mechanisms to enable the self-assembly process to create target structures are required to advance these two fields. Such mechanisms include the design of components (e.g., shape and bonding domains), the design of environments (e.g., stable versus variable conditions), and the design of interactions (e.g., using constraints to dictate component-to-component interactions). The following self-assembly models are provided to review the theoretical advances in staged construction.

Erik Winfree created the abstract tile assembly model (aTAM) to investigate the process of self-assembly [34]. The aim of the aTAM was to link computation with self-assembly by modeling pseudocrystalline growth. The aTAM provides a theoretical framework to investigate the step-by-step process of the self-assembly of 2D DNA tiles in a square lattice environment to create a target structure, algorithmic self-assembly [34]. The aTAM has been used to investigate the algorithmic construction of structures, such as a fully connected n × n square (). The construction of a square is problematic, as assembling tiles must be coordinated to prevent the occurrence of holes.

In the aTAM, a tile type is defined by the bonding domains on the north, west, south, and east edges of a tile. At least one seed tile must be specified to start the self-assembly process. Tiles cannot be rotated or reflected. No more than one tile type can be used at an assembly location in the growing structure. Tile types are in infinite supply, of equal concentration, in the model. All tiles are added to the same environment (one-pot mixture). A seed tile is first placed in the square lattice environment, and additional tiles are added one at a time if the bonding constraints are satisfied.

In the theoretical context of the aTAM, the temperature of the system dictates which tiles can self-assemble together. Tiles can only self-assemble together if the bonding domains meet or exceed the temperature parameter, . In the aTAM, each bonding domain is given a strength value . At Φ1 (temperature 1), any bonding strength of 1 or greater is enough by itself to assemble two tiles together. At Φ2 (temperature 2) a single strength 1 bond is insufficient to assemble two tiles together. Therefore, at least two strength 2 bonds are required to assemble a tile to the growing structure. As a result, temperature dictates cooperative bonding.

However, the physical realization of Φ2 tile-based systems has proven to be difficult (e.g., partial attachments occur between components), and continued investigations into the characteristics of self-assembly at Φ1 are required [23]. Two extensions to the aTAM at Φ1 include the staged tile assembly model (sTAM) and the restricted glue tile assembly model (rgTAM).

The sTAM addresses the challenge of using a set of fixed components with limited bonding mechanisms (i.e., DNA tiles) by dividing the self-assembly process into time intervals based on the temporal aspects of conducting laboratory experiments [12]. Tiles can be added to, or removed from, a set of environments in the sTAM. These abstract operations of adding and removing tiles in the sTAM are based on the laboratory operations of adding and filtering DNA-based components to and from solutions that can be mixed together. As with the aTAM, the sTAM has been used to investigate the algorithmic construction of fully connected n × n squares. The sTAM has shown an algorithmic efficiency with minimal tile sets and bonding mechanisms (not requiring cooperative bonding, i.e., Φ1) in the construction of fully connected squares. This efficiency is due to staging, and is an advantage over the aTAM itself, which relies on cooperative bonding at Φ2 [29] or other extensions to the aTAM by varying the temperature [18] or by varying the concentration of tiles [1, 13].

In contrast to the sTAM, which relies on staging to enable the self-assembly process at Φ1, the rgTAM relies on restricted glue strengths (i.e., bond strengths) [23]. The rgTAM is constrained to glue strengths of −1, 0, and 1 (i.e., repulsive, neutral, and attractive—neutral meaning where no assembly can occur). Conflicting glue types are not able to interact with one another (i.e., the glue interaction matrix is diagonal). The term “restricted” in the name “rgTAM” is used to distinguish it from other extensions to the aTAM that use negative glue types and nondiagonal glue interaction matrices. The physical creation of DNA tiles with repulsive interactions (negative glues) can be achieved by attaching magnetic particles to DNA [19, 27]. By using repulsive, neutral, and attractive glue strengths, the rgTAM can exploit the concept of a zigzag tile assembly system [10]. Zigzag systems are those that grow horizontally one row at a time, by alternating left-to-right and right-to-left growth in only one direction (e.g., north and not south). The combination of repulsive, neutral, and attractive glue strengths is used to enable the self-assembly sequence of a target structure, such as a fully connected n × n square, by growing the structure temporally and spatially one row at a time. In contrast, an n × n square can be achieved using the aTAM by (1) using a seed tile as one of the corners of the square, (2) using bonds of strength 2, extending from the seed tile, to define the length and width of the square, and (3) using cooperative bonds at Φ2 to complete the interior of the square.

The aTAM has also been extended to 3D using cube-based tiles [39]. The advantage of 3D DNA tiles has been shown in computing certain types of functions by self-assembly [26]. However, the physical creation of 3D DNA tiles does not appear to have been achieved to date.

2.3 Physical Staged Self-Assembling Systems

Situated development is another method of investigating staged construction, where artificial evolution has been used to evolve the assembly plan of a structure [28]. Based on 3D printing, assembly plans were evolved using permanent and temporary components which were “dropped” into an environment. Temporary components act as scaffolding and can be removed (representing how support material can be removed in 3D printing).

In contrast to [12, 23, 28], two examples of physical staged self-assembly include the use of environment templates [38] and component concentrations [15], both of which enable self-assembly in a two-stage process. Templates were used to self-assemble spherical beads into substructures with specific patterns (e.g., linear, triangular, and hexagonal shapes), after which the substructures were placed in a new environment to form various lattice structures. Three-point star motif components were used to self-assemble tetrahedrons, dodecahedrons, and buckyballs by controlling the motif length and concentration of components. Despite this work, there is little (if any) literature that describes the use of morphological information to stage the self-assembly process.

3 The Three-Level Approach and Staging

Although there has been progress in the design of self-assembling systems from an algorithmic perspective, the creation of physical self-assembling systems remains challenging. We developed the three-level approach as an alternative bottom-up method to design self-assembling systems [8]. The three phases in our approach include: (1) definition of the rule set, (2) virtual execution of the rule set, and (3) physical realization of the rule set (Figure 1). The motivation behind the three-level approach is in finding the fundamental information structures and rules that enable self-assembly in theory (level 1), testing and refining those rules through simulation (level 2), and testing and refining those rules through physical experiments (level 3) [8]. The three-level approach is inherently bottom-up, as it is possible to directly map a set of self-assembly rules to a physical system.

Figure 1. 

The three-level approach to designing self-assembling systems.

Figure 1. 

The three-level approach to designing self-assembling systems.

The three-level approach provides a high-level description for designing self-assembling systems via physically encoded information [2, 3, 5–8]. In this section, the three-level approach is extended to include our staging strategy by exploiting component physical features (morphological information) at different time intervals during the self-assembly process [7]. At Level 1, a new self-assembly rule is introduced to specify which components are present at a particular time interval. To accommodate this new rule type, an extension to a tile-based self-assembly model is provided at level 2. In contrast to the sTAM and the rgTAM, this new staged self-assembly tile-based model is better suited to the type of physical self-assembling systems we are interested in, because it incorporates parallel construction, component rotation, and component concentration, and uses either 2D or 3D tiles. Physical component features exploited by our staging strategy are described at level 3. Finally, an analytical comparison of the non-staged versus the staged process is provided to justify applying our three-level approach to designing self-assembling systems.

Detailed examples of designing staged self-assembling systems using the three-level approach are used to achieve the objectives of the staging strategy presented, which are to demonstrate how morphological information can be used to further reduce errors (Section 4.1, Two-Dimensional Experiment), exploit rotational properties (Section 4.2, Three-Dimensional Experiments), and create more complex structures that would not otherwise be possible (Section 5, Leveraging Limited Rule Sets).

3.1 Level 1: Definition of Rule Set

A system is described by three categories of self-assembly rules: component, environment, and system. These rules are defined in the context of component movement spatially in 2D or 3D.

Component rules specify shape and information. Conceptually similar to DNA tiles, components are either squares (2D) or cubes (3D). Each edge or face of a component serves as an information location (Figure 2), in either a four-point (top-left-bottom-right) or a six-point arrangement (top-left-bottom-right-front-back). Information is represented by a capital letter (A to H for 2D components, and I to T for 3D components). The letters represent linear sequences in the 2D case, and planar formations in the 3D case. A subscript (1 to 4) is used with each capital letter (e.g., N4) to indicate the orientation of the planar information in relation to the four edges of a 3D component's face (Figure 2). The dash symbol (-) represents a neutral site (where no assembly information is present). The spatial relationship of a component's information defines its type (Figure 2).

Figure 2. 

2D and 3D component spatial information relationship ((a) and (e), respectively), an example of information orientation on a 3D component's face (f ), and example 2D component types (where (c) and (d) are of the same type under planar rotation and are distinguished from (b)).

Figure 2. 

2D and 3D component spatial information relationship ((a) and (e), respectively), an example of information orientation on a 3D component's face (f ), and example 2D component types (where (c) and (d) are of the same type under planar rotation and are distinguished from (b)).

Environment rules specify environmental conditions such as temperature (Φ) and boundary constraints. An assembly protocol must at least meet the temperature requirement for assembly bonds to occur. The boundary confines components to the environment. Components are permitted to translate and rotate in 2D and 3D systems. Furthermore, components can be reflected in 3D systems.

System rules specify the frequency of component types in each time interval (ψ), and two interaction rules (fits and breaks). Time intervals indicate when components are added to a single environment (e.g., ψ0; we use subscripts 0 to m, where and 0 indicates the start of the self-assembly process). If two complementary pieces of information come into contact (e.g., A fits B), it will cause them to assemble. This rule type is commutative (e.g., if A fits B, then B fits A). Furthermore, fits rules encapsulate component-to-component rotational interactions in 3D systems. A subscript (360, 180, or 90) is used to represent whether the faces of complementary 3D components can fit together in four ways, two ways, or only one way, respectively (e.g., M fits180 N). Complementary pieces of information self-assemble at Φ1. If two assembled pieces of information experience at least Φ2, then their assembly breaks. The system rules used are provided in Table 1 (2D) and Table 2 (3D).

Table 1. 

2D interaction rules (fits and breaks; ‘→’ transition, ‘+’ assembly, ‘;’ disassembly, and ‘Φ2’ temperature 2), where A–H represent component information.

Fits rule
Breaks rule
A fits B → A + B Φ2 breaks A + B → A ; B 
C fits D → C + D Φ2 breaks C + D → C ; D 
E fits F → E + F Φ2 breaks E + F → E ; F 
G fits H → G + H Φ2 breaks G + H → G ; H 
Fits rule
Breaks rule
A fits B → A + B Φ2 breaks A + B → A ; B 
C fits D → C + D Φ2 breaks C + D → C ; D 
E fits F → E + F Φ2 breaks E + F → E ; F 
G fits H → G + H Φ2 breaks G + H → G ; H 
Table 2. 

3D interaction rules (fits and breaks; ‘→’ transition, ‘+’ assembly, ‘;’ disassembly, and ‘Φ2’ temperature 2), where I–T represent component information and 360, 180, and 90 represent rotational interactions.

Fits rule
Breaks rule
I fits360 J → I + J Φ2 breaks I + J → I ; J 
K fits360 L → K + L Φ2 breaks K + L → K ; L 
M fits180 N → M + N Φ2 breaks M + N → M ; N 
O fits90 P → O + P Φ2 breaks O + P → O ; P 
Q fits90 R → Q + R Φ2 breaks Q + R → Q ; R 
S fits90 T → S + T Φ2 breaks S + T → S ; T 
Fits rule
Breaks rule
I fits360 J → I + J Φ2 breaks I + J → I ; J 
K fits360 L → K + L Φ2 breaks K + L → K ; L 
M fits180 N → M + N Φ2 breaks M + N → M ; N 
O fits90 P → O + P Φ2 breaks O + P → O ; P 
Q fits90 R → Q + R Φ2 breaks Q + R → Q ; R 
S fits90 T → S + T Φ2 breaks S + T → S ; T 

3.2 Level 2: Virtual Execution of Rule Set

At level 2, a self-assembly rule set is mapped to an abstract tile-based model for computational efficient evaluation, and is used to determine whether physical evaluation of a self-assembly rule set is applicable at level 3. We extend the concurrent tile assembly model (cTAM [6, 8]) to incorporate staging. In contrast to the aTAM, the cTAM is better suited to the type of self-assembling systems used here in allowing multiple substructures to self-assemble concurrently, not using seed tiles, permitting more than one tile type to be used at an assembly location, and requiring all tiles to be in the same one-pot-mixture environment. The extended cTAMs are referred to as the 2D and 3D staged concurrent tile assembly models (2DscTAM and 3DscTAM). Unlike the aTAM and sTAM, components are permitted to translate and rotate in both the 2DscTAM and the 3DscTAM, but be reflected only in the 3DscTAM. Components can only be added to the one-pot-mixture environment at each stage in the 2DscTAM and the 3DscTAM, unlike the sTAM, where components can also be removed at each stage.

The input into the 2DscTAM and the 3DscTAM is the number of time intervals, and the multiset of components in each interval (type and frequency). At the start of each time interval, the components corresponding to the current time interval are added to the environment (Figure 3). A single assembly operation is applied during a time interval, initialized by selecting a single tile or substructure with an open assembly location at random. If no other tile or substructure has an open complementary information location, then the location on the first tile or substructure is labeled unmatchable. If there are tiles or substructures with open complementary information locations, all those tiles or substructures are put in an assembly candidate list. From the assembly candidate list, tiles or substructures are selected at random until a tile or substructure can be added. If no such tile or substructure can be added, due to an assembly violation (Figure 4), then the location is labeled unmatchable. If a tile or substructure can be added, the open assembly locations on the two tiles or substructures are updated and labeled match (all applicable assembly locations, including their rotational properties in the 3D case, must match when adding two substructures). This process repeats until all assembly locations are set to either match or unmatchable. At the end of a time interval, the resulting structures are placed in a single grid environment to determine whether boundary violations occur. Before starting the next time interval, all unmatchable information locations are reset. The algorithm repeats, and halts when all time intervals have been completed in sequence. Pseudocode for this algorithm is provided in [2].

Figure 3. 

Example 2DscTAM steps for constructing a 3 × 3 square using two time intervals.

Figure 3. 

Example 2DscTAM steps for constructing a 3 × 3 square using two time intervals.

Figure 4. 

2DscTAM examples of assembly violations (left) and no assembly path (right, meaning that no sequence of translation and rotation steps can be used to move the smaller substructure into the interior of the larger substructure for assembly to occur).

Figure 4. 

2DscTAM examples of assembly violations (left) and no assembly path (right, meaning that no sequence of translation and rotation steps can be used to move the smaller substructure into the interior of the larger substructure for assembly to occur).

An added constraint to the 3DscTAM is that substructures (with three or more components) cannot assemble together. This constraint represents observations in preliminary physical experiments conducted by the authors [8].

3.3 Level 3: Physical Realization of Rule Set

Components are physically realized using 3D printing, at level 3. Both 2D and 3D components are defined by their design space (set of physically feasible designs, Figures 5 and 6). The design space is a combination of a shape space and an assembly protocol space. The component design space used here results in component interactions considered as a physical realization of the component interactions in the rgTAM, to achieve bond strengths of absolute value 1 (see Section 2.2, Algorithmic Self-Assembly). Magnetism is used to create attractive and repulsive component interactions (positive and negative bond/glue strengths). Repulsive tile interactions are not incorporated into the 2DscTAM and the 3DscTAM, as the combination of component shape and magnetic-bit patterns is used to reduce or prevent conflicting component information from interacting (i.e., the bond/glue interaction matrix is diagonal) in physical systems.

Figure 5. 

2D component specifications (dimensions in millimeters).

Figure 5. 

2D component specifications (dimensions in millimeters).

Figure 6. 

3D component specifications (dimensions in millimeters), where solid black circles represent magnetic north and outlined black circles represent magnetic south in the 5-magnetic-bit patterns (the orientations of the planar magnetic formations are shown in reference to edge 1, e.g., Q1; see Section 3.1).

Figure 6. 

3D component specifications (dimensions in millimeters), where solid black circles represent magnetic north and outlined black circles represent magnetic south in the 5-magnetic-bit patterns (the orientations of the planar magnetic formations are shown in reference to edge 1, e.g., Q1; see Section 3.1).

For both 2D and 3D components, a key-lock-neutral concept defines the shape space. A linear 3-magnetic-bit and a planar 5-magnetic-bit encoding scheme define the assembly protocol space for 2D and 3D components, respectively. Magnets are recessed within the edges or faces of 2D and 3D components, respectively. The resulting air gap between the magnet and component surface allows for adjustable component interactions and selective bonding [33]. Although Mayashita et al. [22] investigated how component shape and magnetic bonding affect the self-assembly process, they did not consider this morphological information in the context of staged self-assembly.

Here, lock-to-lock interactions can never occur, due to the shape of the components. This shape characteristic is influential in assigning 3-magnetic-bit and 5-magnetic-bit patterns to keys and locks. One magnet is placed in each position associated with a key, and two magnets are placed in each position associated with a lock. Strong bonding is ensured for key-to-lock interactions, and weak bonding for key-to-key interactions. The likelihood of weak bonding can be reduced with an appropriate physical temperature setting.

The four pairs of complementary 3-magnetic-bit patterns can be optimally assigned to keys and locks to reduce assembly errors, as any key-to-lock error is at worst a one-out-of-three match (Table 3). Since this is not above a 50% match, bonding will not occur. The six pairs of unique complementary pairs of 5-magnetic-bit patterns (accounting for planar rotation of a component's face) cannot be optimally assigned to keys and locks to reduce assembly errors. In this case, optimal assignment is considered with respect to which encodings are included to construct a target structure in the experiments (i.e., prevent conflicts between 360° and 90° patterns). Table 4 lists an assignment of 5-magnetic-bit patterns to keys and locks that are used in the experiments in the following section. It should be noted that these six pairs of patterns encapsulate rotational information for 3D component-to-component interactions, where two pairs encapsulate 360°, one pair encapsulates 180°, and three pairs encapsulate 90° rotational interactions. The 90° patterns have the potential for self-errors between complementary pairs, that is, for a 3-out-of-5 match. In 3D systems, a physical temperature to break 3-out-of-5 matches while maintaining 5-out-of-5 matches between 90° 5-magnetic-bit patterns is striven for.

Table 3. 

Key and lock designations for the 3-magnetic-bit patterns (zero and one represent magnetic south and north, respectively).

Key/lock
3 magnetic bits
Label
Fits rule
Breaks rule
Lock 000 A fits B → A + B Φ2 breaks A + B → A ; B 
Lock 110 C fits D → C + D Φ2 breaks C + D → C ; D 
Lock 011 E fits F → E + F Φ2 breaks E + F → E ; F 
Lock 101 G fits H → G + H Φ2 breaks G + H → G ; H 
Key 111 B fits A → B + A Φ2 breaks B + A → B ; A 
Key 001 D fits C → D + C Φ2 breaks D + C → D ; C 
Key 100 F fits E → F + E Φ2 breaks F + E → F ; E 
Key 010 H fits G → H + G Φ2 breaks H + G → H ; G 
Key/lock
3 magnetic bits
Label
Fits rule
Breaks rule
Lock 000 A fits B → A + B Φ2 breaks A + B → A ; B 
Lock 110 C fits D → C + D Φ2 breaks C + D → C ; D 
Lock 011 E fits F → E + F Φ2 breaks E + F → E ; F 
Lock 101 G fits H → G + H Φ2 breaks G + H → G ; H 
Key 111 B fits A → B + A Φ2 breaks B + A → B ; A 
Key 001 D fits C → D + C Φ2 breaks D + C → D ; C 
Key 100 F fits E → F + E Φ2 breaks F + E → F ; E 
Key 010 H fits G → H + G Φ2 breaks H + G → H ; G 
Table 4. 

Key and lock designations for the 5-magnetic-bit patterns (zero and one represent magnetic south and north, respectively, and magnetic information is represented linearly from left to right as: center, bottom right, bottom left, top left, and top right).

Key/lock
5 magnetic bits
Label
Fits rule
Breaks rule
Lock 00000 I fits360 J → I + J Φ2 breaks I + J → I ; J 
Lock 10000 K fits360 L → K + L Φ2 breaks K + L → K ; L 
Lock 01010 M fits180 N → M + N Φ2 breaks M + N → M ; N 
Lock 10011 P fits90 O → P + O Φ2 breaks P + O → P ; O 
Lock 00111 R fits90 Q → R + Q Φ2 breaks R + Q → R ; Q 
Lock 10111 T fits90 S → T + S Φ2 breaks T + S → T ; S 
Key 11111 J fits360 I → J + I Φ2 breaks J + I → J ; I 
Key 01111 L fits360 K → L + K Φ2 breaks L + K → L ; K 
Key 10101 N fits180 M → N + M Φ2 breaks N + M → N ; M 
Key 01100 O fits90 P → O + P Φ2 breaks O + P → O ; P 
Key 11000 Q fits90 R → Q + R Φ2 breaks Q + R → Q ; R 
Key 01000 S fits90 T → S + T Φ2 breaks S + T → S ; T 
Key/lock
5 magnetic bits
Label
Fits rule
Breaks rule
Lock 00000 I fits360 J → I + J Φ2 breaks I + J → I ; J 
Lock 10000 K fits360 L → K + L Φ2 breaks K + L → K ; L 
Lock 01010 M fits180 N → M + N Φ2 breaks M + N → M ; N 
Lock 10011 P fits90 O → P + O Φ2 breaks P + O → P ; O 
Lock 00111 R fits90 Q → R + Q Φ2 breaks R + Q → R ; Q 
Lock 10111 T fits90 S → T + S Φ2 breaks T + S → T ; S 
Key 11111 J fits360 I → J + I Φ2 breaks J + I → J ; I 
Key 01111 L fits360 K → L + K Φ2 breaks L + K → L ; K 
Key 10101 N fits180 M → N + M Φ2 breaks N + M → N ; M 
Key 01100 O fits90 P → O + P Φ2 breaks O + P → O ; P 
Key 11000 Q fits90 R → Q + R Φ2 breaks Q + R → Q ; R 
Key 01000 S fits90 T → S + T Φ2 breaks S + T → S ; T 

Orbital shakers form the environments for both 2D and 3D components. 2D components are placed on the surface of a tray, and a lid is used to prevent component reflections. 3D components are placed in a jar of mineral oil, to allow components to move freely in 3D space, and prevent oxidation affecting the magnets. The designs for both environments result from earlier experiments conducted by the authors [6, 8]. Detailed specifications of 2D and 3D components, and their respective environments, are provided in [2].

3.4 Non-staged versus Staged Self-Assembly

With the physical component design space used here, there are 1,665 and 53,977,737 unique 2D and 3D component types, respectively [2, 4]. The number of resulting self-assembled structures constructed from these component types depends on the sequence of assembly steps. The following worst-case example is used to quantify the possible number of self-assembled structures to compare the non-staged versus the staged self-assembly process. In this scenario there are two types of 3D components: component type 1 has information I on all its faces, and component type 2 has information J on all its faces. In the non-staged case, a set containing z type 1 components and z type 2 components () would result in at most 2z! structures [2]. In the staged case, a set containing z type 1 components and z type 2 components, and number of stages ψm (, where m = 0 is used to start the staged self-assembly process; see Section 3.1) would result in at most (2z!)m+1 structures. Consequently, it is not practical to test all the possible structures resulting from a large component set and a large number of stages. Therefore, it is justified to map a set of level 1 rules to the 2DscTAM or 3DscTAM at level 2 for computationally efficient evaluation, and to determine whether a physical evaluation at level 3 is applicable. The numbers of components and stages used in the experiments presented in the next section are small. However, it is our contention that our tile-based models are beneficial in evaluating a set of self-assembly rules, as it is difficult to determine the resulting structures given the number of component types, rotations, and interacting substructures in parallel.

4 Experiments and Results

Four experiments were conducted to test our approach to staging the self-assembly process using morphological information. The purpose of these experiments was to demonstrate, as proof of concept, that staging can be used to further reduce errors during the self-assembly process and exploit rotational properties of component-to-component interactions in the creation of closed target structures (“closed” refers to structures with defined boundaries [33]). A target structure was assigned to each experiment (one 2D and three 3D experiments, Figure 7).

Figure 7. 

The four target structures for the experiments.

Figure 7. 

The four target structures for the experiments.

Due to the fixed components with limited bonding mechanisms used in this work, the staged self-assembly process is an additive process, where a target structure is constructed from the inside out. At each time interval, new components are added to the one-pot-mixture environment. Staged component sets were created using a combination of top-down decomposition and bottom-up trial and error. The morphology of a target structure was used to identify the connectivity (number and position of neighboring components) of each component, and to determine the number of unique components (based on connectivity) and their frequency. The connectivity of each unique component established the symmetric and asymmetric information required, and was used to assign component information to create each component type. Component types that have the potential to lead to conflicts are separated into different time intervals. In addition, component types that can increase the symmetry of substructures at Φ1 (i.e., without the use of cooperative bonding) are also separated into different time intervals.

In the proof-of-concept systems presented here, fully connected target structures are desired, meaning that neighboring components must be assembled with one another (e.g., no neighboring neutral shapes or neighboring neutral and lock shapes). Although fully connected self-assembled structures are not required in general, this constraint is used here for reasons of structural rigidity. Another constraint placed on the systems presented here is that each time interval displays a one-component growth of substructures after the initial stage. A component (tile) used in the 2DscTAM or 3DscTAM and physical systems can be considered as one unit in size. The implication of this stepwise construction is that the existing substructure can only grow by at most one component (unit) along all axes in Cartesian space.

In the experiments presented in this section, the self-assembly process is staged (divided) into two time intervals, during which components are added by hand to a one-pot-mixture environment. Component physical features, such as key and lock shapes and magnetic-bit patterns, are morphological information. The independent variable is the use of the two time intervals. The dependent variable is the resulting self-assembled structures. Enough components are supplied to create one 2D target structure and two 3D structures (due to the boundary constraints of the environment). Ten trials are run for each experiment. A virtual trial (level 2) is evaluated as successful if all the potential target structures are achieved. A physical trial (level 3) is evaluated as successful if at least one target structure is achieved. The staging strategies and level 1 rules were designed by the authors. 2D and 3D experimental procedures and results are provided in terms of the three-level approach.

4.1 Two-Dimensional Experiment

The staging strategy for creating the 2D 3 × 3 square target structure is to construct the center and edges of the square in the first time interval, and construct the corners of the square in the second time interval (Figure 8). In the first time interval, potential errors between the edge components can be reduced by appropriate selection of 3-magnetic-bit patterns and the use of lock shapes to assemble to the center component. The morphology of the substructure after the first time interval has corner features that can reduce assembly errors with the use of corner components that use only lock assembly shapes. The neutral edges of the corner components effectively block a corner component from assembling to the substructure in an improper orientation (Figure 8).

Figure 8. 

Staging strategy for target structure I, and error prevention due to shape and proper 3-magnetic-bit pattern selection (e.g., avoiding magnetic repulsion configuration). Solid black circles represent magnetic north, and open black circles represent magnetic south.

Figure 8. 

Staging strategy for target structure I, and error prevention due to shape and proper 3-magnetic-bit pattern selection (e.g., avoiding magnetic repulsion configuration). Solid black circles represent magnetic north, and open black circles represent magnetic south.

4.1.1 Level 1: Rule Set for 2D Experiment

Table 5 provides the component rules. The control group represents components that were not divided into time intervals (non-staged). The experimental group used the same components, but divided them into two time intervals (staged). The environment used Φ1. Interaction rules from Table 3 were applicable to both groups.

Table 5. 

Staged component set for the 2D experiment (represented as “ψ # × (Top, Left, Bottom, Right),” where ψ identifies the time interval, # represents quantity, and the directions refer to component information).

Target structure
Staged component set
ψ0 {1 × (D,D,D,D), 4 × (B,—,B,C)} 
ψ1 {4 × (—,A,A,—)} 
Target structure
Staged component set
ψ0 {1 × (D,D,D,D), 4 × (B,—,B,C)} 
ψ1 {4 × (—,A,A,—)} 

4.1.2 Level 2: Experimental Setup for 2D Experiment

The components from Table 5 were mapped to an abstract representation for the 2DscTAM. Each component's shape was a unit square. The size of the environment was 10 × 10 units (as a representation of width × depth, and the ratio between component and environment size). A different random seed was used to initialize the 2DscTAM for each trial.

4.1.3 Level 2: Experimental Results for 2D Experiment

The staged components successfully created one target structure in each of the 10 trials. None of the non-staged components was able to create one target structure. The unsuccessful non-staged trials either resulted in a set of substructures (due to edge and corner components assembling in incorrect orientations), or the creation of a 3 × 3 open square. The results at level 2 were analyzed using Fisher's exact test (one-sided) for binary data [11]. The results are statistically significant with a p-value of 0.

4.1.4 Level 3: Experimental Setup for 2D Experiment

A level 3 translation was preformed for both the staged and non-staged component sets (to observe the physical results of non-staged component sets). Components were mapped following Table 5.

An Eden 333 Polyjet 3D printer was used to fabricate the components from Vero Grey resin. Neodymium (NdFeB) disk magnets (1/16 × 1/32 in., diameter × radius; grade N50) were inserted into the components. Blue (red) paint for north (south) marked the magnets. Further information on fabricating the components is provided in [2].

The environment size was mapped in accordance with the base component's size, to specify the dimensions of the circular tray environment. The tray was fabricated using a Dimensions Elite 3D printer, using ABS plastic (the sparse-fill option was used to create a rough surface texture). The outer radius of the tray is 135 mm and the inner radius is 125 mm, while the outer wall height is 9 mm and the inner wall height is 6 mm. The tray was mounted to a Maxi Mix II vortex mixer (using a tray mounting bracket, also fabricated using the Dimensions 3D printer). A tray lid was cut using a Trotec Speedy 300 Laser Engraver laser cutting machine, using a 2-mm clear acrylic sheet. The tray lid was secured to the tray using polycarbonate screws and wing nuts. Details regarding the construction of the environment are provided in [2].

Each physical trial followed seven steps: (1) Set the speed control on the vortex mixer to 1,050 rpm. This speed created an appropriate shaking level (environment temperature) to maintain fits rules, and to mostly break partially matched magnetic patterns. (2) Secure the mixer to a table, using a 3-in. C clamp and six hex nuts (to help secure the C clamp to the back of the mixer). (3) Randomly place components on the surface of the tray (trying to ensure that complementary bonding locations on the components are not in line with each other). (4) Secure the tray lid. (5) Run the mixer for 20 min for a non-staged trial, or for two 10-min intervals for a staged trial. (6) Turn the mixer off. (7) Record the state of the system, observing the number of target structures created, the number of matching errors (between conflicting physical information, where no fits rule is applicable), and the number of assembly errors (partial attachment between complementary physical information, where a fits rule is applicable).

4.1.5 Level 3: Experimental Results for 2D Experiment

The level 3 results are provided in Table 6, with an example of the end of each time interval of a successful trial (Figure 9). For both component groups, no matching and assembly errors were observed in the 10 trials. Only partial structures, and no open 3 × 3 squares, were observed at the conclusion of the non-staged trials. Using Fisher's exact test, this experiment is statistically significant at the 0.01 level (i.e., there is a 99% certainty the results are not due to chance). The results of this experiment demonstrate how staging can be used to reduce errors by using the morphology of the substructure resulting from the first time interval, to reduce the potential for the occurrence of a hole in the target structure, and to reduce the number of unique types by leveraging symmetry in the context of Φ1 self-assembly.

Table 6. 

The numbers of successful and unsuccessful trials for the staged and non-staged trials for the 2D experiment, with corresponding p-value using Fisher's exact test (one-sided) for analyzing binary data.

Target structure
Group
Successful
Unsuccessful
p-value
Staged  
Non-staged 10 
Target structure
Group
Successful
Unsuccessful
p-value
Staged  
Non-staged 10 
Figure 9. 

Successful example trial for target structure I.

Figure 9. 

Successful example trial for target structure I.

4.2 Three-Dimensional Experiments

The three 3D target structures have a three-component common core structure, and vary in the number of periphery components (increasing from two to three to four; see Figure 7). The core structure requires two specialized 90° bonds, whereas the perimeter components only require general 360° bonds. As observed by the authors in preliminary 3D experiments, substructures consisting of at least three components are not able to assemble together [8]. Given that the occurrence of general 360° bonds is more likely than that of specialized 90° bonds, the staging strategy for creating the three 3D target structures is to construct the core substructure in the first time interval, and construct the periphery substructures in the second time interval (Figure 10). The first time interval exploits the specialized component rotational information. Lock shapes for the 360° bonds are used as part of the morphology of the components in the first time interval, to reduce potential matching errors between specialized and general bonds. Furthermore, the resulting morphologies of the resulting core substructures at the end of the second time interval consist only of neutral and lock shapes, preventing assembly between the core substructures.

Figure 10. 

Staging strategy for target structure III (applicable to target structures II and IV).

Figure 10. 

Staging strategy for target structure III (applicable to target structures II and IV).

4.2.1 Level 1: Rule Set for 3D Experiments

The component rules for the 3D experiments are provided in Table 7. Control groups and experimental groups represent non-staged and staged (using two time intervals) component sets, respectively. The environment used Φ1. The interaction rules from Table 4 applied to both groups.

Table 7. 

Staged component set for the 3D experiments (represented as “ψ # × (Top, Left, Bottom, Right, Front, Back),” where ψ identifies the time interval, # represents quantity, the directions refer to component information, and the component subscripts represent the orientation of the information on a 3D component's face; see Section 3.1 and Figure 6).

Target structure
Staged component set
II ψ0 {2 × (—,—,O3,—,O1,—), 4 × (—,I1,—,—,P1,—)} 
ψ1 {4 × (J1,—,—,—,—,—)} 
III ψ0 {2 × (—,Q1,—,Q1,K1,—), 4 × (—,—,—,K1,R1,—)} 
ψ1 {6 × (L1,—,—,—,—,—)} 
IV ψ0 {2 × (T1,—,T4,—,—,—), 4 × (—,—,I1,I1,S1,—)} 
ψ1 {8 × (J1,—,—,—,—,—)} 
Target structure
Staged component set
II ψ0 {2 × (—,—,O3,—,O1,—), 4 × (—,I1,—,—,P1,—)} 
ψ1 {4 × (J1,—,—,—,—,—)} 
III ψ0 {2 × (—,Q1,—,Q1,K1,—), 4 × (—,—,—,K1,R1,—)} 
ψ1 {6 × (L1,—,—,—,—,—)} 
IV ψ0 {2 × (T1,—,T4,—,—,—), 4 × (—,—,I1,I1,S1,—)} 
ψ1 {8 × (J1,—,—,—,—,—)} 

All three pairs of 90° 5-magnetic-bit patterns were successful in previous physical experiments to create target structures [8]. No particular pair of 90° patterns was found to be superior. Therefore, one pair of 90° 5-magnetic-bit patterns was assigned by the authors to each set of components corresponding to one of the three 3D experiments. After each 90° pattern was assigned, the corresponding 360° pattern with minimal potential error interaction (I–J or L–K) was assigned to each set of components (Table 4).

4.2.2 Level 2: Experimental Setup for 3D Experiments

The components from Table 7 were mapped to an abstract representation for the 3DscTAM. A component's base shape was a unit cube. The size of the environment was 4 × 4 × 4 units (representing width × depth × height, and the ratio between component and environment size). A different random seed was used to initialize the 3DscTAM for each trial.

4.2.3 Level 2: Experimental Results for 3D Experiments

The staged components, for each experiment, successfully created two target structures in each of the 10 trials, whereas the non-staged components were not able to create a target structure. As expected, the unsuccessful non-staged components resulted in substructures consisting of three components (favoring assemblies with 360° bonds) or two components. The results at level 2 are statistically significant with a p-value of 0 using Fisher's exact test for binary data.

4.2.4 Level 3: Experimental Setup for 3D Experiments

As with the 2D experiment, a level 3 translation was performed for both staged and non-staged components (to observe the physical results of non-staged components). Components were mapped following Table 7, and were fabricated using a similar procedure to that for the 2D components (with the addition of color paints to represent rotational interaction information, as in Figure 6; see [2] for details).

Clear glass 500 mL wide-mouth jars with rubber-lined lids were used to contain components (91 × 95 mm, diameter × height). A Trotec Speedy 300 laser engraver was used to construct the parts, using 3-mm acrylic sheet, for the jar rack. The rack was assembled using adhesive, screws, and hex nuts. The jar rack was placed on a New Brunswick Scientific Excella E1 platform shaker. An amount of 325 mL of Rogier Pharma light mineral oil was measured using a graduated cylinder and poured into the jars (one for each experiment). Full instructions for constructing the environment are provided in [2].

Each physical trial followed six steps: (1) Place three jars of mineral oil on the jar rack. (2) Randomly place the components for each experiment in the appropriate jar. (3) Secure the jar lids. (4) Turn the shaker on by setting the speed to 32.5 rpm. (5) Run the shaker for 40 min for a non-staged trial, or for two 20-min intervals for a staged trial. (6) Record the state of each system, observing: the number of target structures created, the number of matching errors, the number of assembly errors, and the number of rotation errors (between complementary components).

4.2.5 Level 3: Experimental Results for 3D Experiments

The 3D level 3 results are provided in Table 8, along with examples of the end of each time interval of a successful staged trial (Figure 11). For each experiment, no matching or assembly errors were observed in the 10 trials. Rotational errors were observed in each staged experiment (Figure 12). Using Fisher's exact test, the first two 3D experiments are statistically significant at the 0.05 level, and the third experiment was statistically significant at the 0.50 level (i.e., there is a 95% and a 50% certainty the results are not due to chance). Even though one successful staged trial was observed with the third 3D experiment, we do not consider the result statistically relevant. The results of these experiments demonstrate how specialized component rotational information can be exploited using staging. These results also demonstrate how difficult it is to achieve an appropriate physical environment temperature in practice.

Table 8. 

The numbers of successful and unsuccessful staged and non-staged trials for the 3D experiment, with corresponding p-values using Fisher's exact test (one-sided) for analyzing binary data.

Target structure
Group
Successful
Unsuccessful
p-value
II Staged  
Non-staged 10 
III Staged  
Non-staged 10 
IV Staged  
Non-staged 10 
Target structure
Group
Successful
Unsuccessful
p-value
II Staged  
Non-staged 10 
III Staged  
Non-staged 10 
IV Staged  
Non-staged 10 
Figure 11. 

Successful example trials for target structures II, III, and IV.

Figure 11. 

Successful example trials for target structures II, III, and IV.

Figure 12. 

Rotational errors at the end of each 3D trial (TR1–TR10).

Figure 12. 

Rotational errors at the end of each 3D trial (TR1–TR10).

4.3 Discussion

Four experiments were conducted to demonstrate our morphological-information-based staging strategy. At level 2, all of the staged component sets were able to achieve their respective target structures, whereas none of the non-staged components were able to. All the staged component sets, except for the third 3D experiment, were able to successfully construct their respective target structures at a statistically significant level (with 99% and 95% confidence for the 2D and 3D experiments), at level 3.

In addition to the quantitative observations for conducting statistical significance tests, qualitative observations were recorded for the 2D and 3D experiments. Staging has consequences for the self-repairing properties of a system. Although we observed the 2D 3 × 3 square to be able to self-repair, this was only within the second stage. One physical target structure was achieved in the third 3D experiment, and we observed in the trials a layering effect of components and substructures that inhibited the self-assembly of this target structure (IV).

Although there is sufficient morphological information in the 2D and 3D component design space and resulting interaction rules to construct all four target structures, the results provide evidence that staged self-assembly can be used to further reduce errors and exploit rotational properties. The next section provides analysis of how staging can be used to construct target structures that would not otherwise be possible due to insufficient component morphological information and interaction rules.

5 Leveraging Limited Rule Sets

We now discuss a method to leverage a limited set of self-assembly rules using staging, since it is difficult to construct components in practice [12]. The following example uses a doglike target structure (Figure 13). This example is used to demonstrate how staged self-assembly can also be used to leverage a limited set of component interaction rules (3D 5-magnetic-bit patterns, Figure 6), and create body plan designs inspired by development in biological systems [36].

Figure 13. 

The dog target structure, showing assembly between neighboring components (solid black circles) and the 90° rotational requirements (dashed lines), where the grouped connections can use the same rotational information due to symmetry (and the four corner components in the body can use the same rotational information, as shown in the top view).

Figure 13. 

The dog target structure, showing assembly between neighboring components (solid black circles) and the 90° rotational requirements (dashed lines), where the grouped connections can use the same rotational information due to symmetry (and the four corner components in the body can use the same rotational information, as shown in the top view).

In this example a fully connected dog structure is desired, and the constraints on the staged self-assembly process are the same as those outlined in the previous section. Component types and their rotational properties are based on the connectivity of the components in the dog target structure. As shown in Figure 13, the dog target structure requires six pairs of 90° rotational patterns. However, only three 90° pairs are present in the 5-magnetic-bit encoding scheme (Figure 6). To overcome this deficiency in the number of 90° rotational patterns, staging can be used to:

  • reintroduce previously used component information at later time intervals,

  • use the morphology of a substructure from the dog target structure (at an intermediate stage during the self-assembly process) where neighboring components are used to emulate a 90° rotational pattern using the 180° rotational pattern, and

  • reduce the number of 90° rotational patterns required when constructing only one single target structure (in contrast to multiple target structures), since the orientation of the overall dog target structure during its construction (i.e., head versus tail facing forward) is not important.

The dog target can use all three types of rotational interaction information in a single structure. Table 9 lists the 5-magnetic-bit patterns assigned to key and lock shapes that are better suited to create the dog target structure. In this designation, the worst possible mismatch error between conflicting component information is a 3-out-of-5 positional match, which is the same as in the worst-case scenario between complementary 90° rotational patterns.

Table 9. 

5-magnetic-bit encoding scheme to create the dog target structure, where the differences between the encoding scheme used for the previous 3D experiments (Table 4) are highlighted in boldface.

Key/lock
Five magnetic bits
Label
Fits rule
Breaks rule
Lock 00000 I fits360 J → I + J Φ2 breaks I + J → I ; J 
Lock 01111 L L fits360 K → L + K Φ2 breaks L + K → L ; K 
Lock 01010 M fits180 N → M + N Φ2 breaks M + N → M ; N 
Lock 01100 O O fits90 P → O + P Φ2 breaks O + P → O ; P 
Lock 11000 Q Q fits90 R → Q + R Φ2 breaks Q + R → Q ; R 
Lock 10111 T fits90 S → T + S Φ2 breaks T + S → T ; S 
Key 11111 J fits360 I → J + I Φ2 breaks J + I → J ; I 
Key 10000 K K fits360 L → K + L Φ2 breaks K + L → K ; L 
Key 10101 N fits180 M → N + M Φ2 breaks N + M → N ; M 
Key 10011 P P fits90 O → P + O Φ2 breaks P + O → P ; O 
Key 00111 R R fits90 Q → R + Q Φ2 breaks R + Q → R ; Q 
Key 01000 S fits90 T → S + T Φ2 breaks S + T → S ; T 
Key/lock
Five magnetic bits
Label
Fits rule
Breaks rule
Lock 00000 I fits360 J → I + J Φ2 breaks I + J → I ; J 
Lock 01111 L L fits360 K → L + K Φ2 breaks L + K → L ; K 
Lock 01010 M fits180 N → M + N Φ2 breaks M + N → M ; N 
Lock 01100 O O fits90 P → O + P Φ2 breaks O + P → O ; P 
Lock 11000 Q Q fits90 R → Q + R Φ2 breaks Q + R → Q ; R 
Lock 10111 T fits90 S → T + S Φ2 breaks T + S → T ; S 
Key 11111 J fits360 I → J + I Φ2 breaks J + I → J ; I 
Key 10000 K K fits360 L → K + L Φ2 breaks K + L → K ; L 
Key 10101 N fits180 M → N + M Φ2 breaks N + M → N ; M 
Key 10011 P P fits90 O → P + O Φ2 breaks P + O → P ; O 
Key 00111 R R fits90 Q → R + Q Φ2 breaks R + Q → R ; Q 
Key 01000 S fits90 T → S + T Φ2 breaks S + T → S ; T 

Using this new 5-magnetic-bit encoding scheme arrangement, a single dog structure can be made using six time intervals (Figure 14). In this case, the time interval ψ1 is used to determine the orientation of the head and tail. The second and third time intervals are used to create the main body and to determine the orientation of the head and neck and the direction of the head. The fourth and fifth intervals are used to build the legs and neck. The sixth interval is used to build the head, feet, and tail. This staged process takes advantage of using similar component types in the same time interval, and shows the benefits of symmetry within a time interval for parallel self-assembly. Furthermore, the rotational information Q used in the time interval ψ0 is reused in the time interval ψ3 (Figure 14).

Figure 14. 

Staging for a single dog target structure.

Figure 14. 

Staging for a single dog target structure.

The disadvantage of the previous staged self-assembly process is that it cannot be used to create multiple dog target structures, as it would then be possible to create dogs with either two heads or two tails in the time interval ψ1. To solve this problem, at least one pair from each of the three 90° rotational patterns is required in the time interval ψ1 (Figure 15). However, this results in an insufficient number of pairs of 90° rotational patterns to create the corner parts of the main body of the dog target structure. The 180° rotational pattern can be used, due to the spatial relationship of the neighboring components to the corner components of the main body substructure of the dog target structure (Figure 15). As with the staged self-assembly process to create a single dog target structure, the staged self-assembly process to create multiple dog target structures also uses six time intervals and uses the rotational information Q in the time interval ψ0 again in the time interval ψ3 (Figure 15).

Figure 15. 

Staging for multiple dog target structures, where the last three time intervals correspond to the last three time intervals used to create a single dog target structure (Figure 14).

Figure 15. 

Staging for multiple dog target structures, where the last three time intervals correspond to the last three time intervals used to create a single dog target structure (Figure 14).

This example shows how staged development of a body plan can be used to leverage a limited set of rules. In particular, component interaction rules can be leveraged by reintroducing previously used component information at later time intervals. The physical construction of this staged dog target structure is one area of future work.

6 Future Work and Applications

The constraints placed on the staged self-assembling systems presented here included one-component growth. As future work, we are considering how and which stages can be compressed, so that each stage does not necessarily have to follow a one-component-growth assembly sequence. We are also investigating self-repair properties between stages.

As qualitatively observed with the 2D experiment, self-repair was exhibited in the second time interval. Further research into features that allow for, and the understanding of the limits to, self-repair between specific stages is required to continue to further develop our approach. For example, although salamanders undergo development through unique stages, they can regrow lost limbs by repeating earlier developmental stages [36].

From our perspective, adding new components by hand at each time interval is justifiable to investigate staging based on morphological components for proof-of-concept purposes. We recognize that this type of external intervention is a disadvantage in moving forward toward more complex self-assembling systems. Creating systems using automated staging is the area of future work we consider to be the most significant. To work toward achieving this goal, we are exploring changing environmental conditions, new types of components, and Turing universal computation based on biological development. The influence of changing environmental conditions over time (e.g., in a staged environment) and its effects on the self-assembly process are being investigated. We are also working on new types of components that are more cell-like, in that they can reconfigure themselves based on temporal and positional information to automate staging. Just as the aTAM links computation with self-assembly as a model of pseudocrystalline growth, we are also investigating new models incorporating these more sophisticated types of changing environmental conditions and adaptable components to link computation to developmental processes to construct artificial life forms.

Nevertheless, we envision our staging strategy being applicable to a variety of applications relying on fixed components, such as the design of nano- and micro-scale structures, circuit design, hybrid 3D printing [16], and DNA computing using self-assembly [32, 37]. Moreover, we envision our staging strategy as an approach to improving the ability of artificial evolution for the creation of more complex physical self-assembling systems, scaling to many interacting components.

7 Conclusions

Staging is an essential part of biological development. In this work we have presented a novel approach to staging the self-assembly process using morphological information. This work involved creating two new staged self-assembly analytical tools, the 2DscTAM and the 3DscTAM, which incorporate parallel construction and component rotation at Φ1 (temperature 1). Furthermore, this work showed how the interplay between component morphological information (shape and magnetic patterns) can be used to reduce assembly errors, exploit rotational properties, and leverage a limited rule set by using staging. We presented four proof-of-concept experiments and analysis of a more sophisticated example using the concept of a body plan to demonstrate that our staging strategy is a viable method for enabling the self-assembly of more complex morphologies not otherwise possible.

Acknowledgments

We would like to thank the two anonymous reviewers for their constructive feedback.

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Author notes

Contact author.

∗∗

Swarm Intelligence Group, Department of Computer Science, University of Paderborn, Zukunftsmeile 1, 33102 Paderborn, Germany. E-mail: navneet.bhalla@upb.de

Department of Computer Science, University College London, Malet Place, London, UK, WC1E 6BT. E-mail: p.bentley@cs.ucl.ac.uk

Department of Biological Sciences, Department of Biochemistry and Molecular Biology, and Department of Computer Science, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada, T2N 1N4. E-mail: pvize@ucalgary.ca

§

Department of Computer Science and Department of Biochemistry and Molecular Biology, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada, T2N 1N4. E-mail: cjacob@ucalgary.ca