Blocked Tile Assembly Model (BlockTAM)

In tile-based models of self-assembly such as the aTAM, all input to a system is provided at the beginning of the process in the form of pre-made seed assemblies and copies of unbound tiles of a fixed set of tile types, and then self-assembly proceeds during the duration of a relatively constant temperature. The Blocked Tile Assembly Model, or BlockTAM, is a mathematical model of tile-based self-assembling systems that allows for additional dynamics and types of input to its systems. The major differences from the aTAM are that a series of temperatures can be specified so that the assembly process proceeds through each of them in sequence, and that so-called blockers, which are components that disable glues when the temperatures are in fixed ranges, can be included with systems. These changes allow for a notable increase in the possible dynamics for BlackTAM systems.

Model Definition

The definition of the Block Tile Assembly Model (BlockTAM) is similar to the multiple temperature model in that it is based off of the notion of the aTAM but with the allowance that the temperature can be changed during the self-assembly process. Essentially, assembly proceeds in temperature phases such that for each, a designated temperature (from a given sequence) is set as the system temperature and assembly proceeds until it is terminal. At that point, the next temperature in the sequence is selected and assembly proceeds (if possible) until it is again terminal. This continues until the temperature sequence is completed. An additional caveat in the BlockTAM is that a set of blockers can be defined, where each is specific to a particular glue on a specified tile. Whenever the temperature of the system is lower than the strength of that glue, it is assumed that the blocker attaches to that glue and deactivates it, effectively making it to be of strength-0 whenever the system temperature is less than the glue's strength. By including blockers, it is possible to allow growth to occur at low temperatures relative to a strong glue without that glue being able to bind. This provides a way to have unique growth at different temperatures and provides a way of supplying input to self-assembling systems, without the need for custom tile types or seed assemblies.

The powerful dynamics enabled by the BlockTAM allow for the creation of systems that are universal for various tasks, needing only the input of specific temperature sequences to drive their assembly along arbitrary paths. Below is an example of a BlockTAM system, followed by brief descriptions of a pair of results, and finally links to a web-based simulator and downloadable examples.

Example BlockTAM system

The following figure shows an example of a BlockTAM system.

An example BlockTAM system.

An illustration of an example BlockTAM system. Part (a) shows the definition of the example BlockTAS. Here, a glue label is represented as the color of the protruding squares on a tile and it's strength is the number of protruding squares. For example, the red tile has a green strength 3 glue on its east side. We represent a strength 6 glue by a long rectangle protruding from the edge of the tile. We represent a blocker as two rectangular slices where one rectangle is the same color as the glue which it blocks and the other rectangle is the color of the tile to which the blocker binds. In this example, we assume that all blockers bind with the same strength as the glue. Part (b) shows the assembly formed by the tile set in a traditional TAS without blockers. The tile set grows a width 2 ribbon with alternating red and blue tile segments. Part (c) shows the equivalent blocked tile assembly system for temperature phase \(0\) of \(\mathcal{B}\) along with its terminal assembly. Note the blue tile is unable to bind to the terminal assembly because the two purple glues are strength \(2\), summing to \(4\) which is less than the binding threshold. Part (d) shows the equivalent blocked tile assembly system for temperature phase \(1\) of \(\mathcal{B}\) along with its terminal assembly. Note the red tile is unable to bind to the terminal assembly because it is blocked (making its strength 0 which is less than the binding threshold).

Survey of Results

Initial results in the BlockTAM have shown how so-called universal tile sets can be designed for certain tasks, such as encoding numbers or making assemblies of given shapes, so that the only input specified to them is provided via a series of temperature changes. Those temperature changes are used to drive the self-assembly processes of BlockTAM systems to achieve the specified tasks. Following is a brief overview of some of these results.

Universal number encoder

For any set of temperatures of size \(\ge 3\) there exists a universal number encoder which is a tile set, cover set, and seed tile such that, for any given input number \(n\), a temperature sequence (over the given set of temperatures) exists such that the resulting BlockTAM system is directed and the unique terminal assembly is a rectangle such that the number \(n\) is encoded in the glues on the north of the assembly. (The base and format of the encoding is dependent upon the size of the temperature set that the universal number encoder was built for.)

Base 10 number 100 encoded by the universal number encoder for temperatures 2,3,4

Universal shape builder

There exists a tile set consisting of 145 tile types, 32 blockers, and a single seed tile type, such that, for any finite connected shape \(S\) a temperature sequence over the temperatures \(\{2,3,4\}\) can be generated so that a BlockTAM system using those components and that temperature sequence will self-assemble \(S\) at scale factor \(6\).

Base macrotiles

Example path self-assembled

(Left) The base macrotiles of the universal shape builder, (Right) An example assembly formed by the universal shape builder

Examples and Simulation Software

The BlockTAM can be simulated by (the currently beta version of) | WebTAS. However, note there are a few important caveats:

• Although WebTAS(beta) supports the BlockTAM by progressing through specified temperature phases, correctly setting the temperature and blockers for each, enforcing the minimum binding threshold for individual tiles based on the temperature, and letting each phase run until assembly is terminal, it does not currently check for melting. That is, in the case of a temperature increase it does not compute the minimum cuts across the binding graph of the assembly to see if any individual tiles or portions have a maximum cut below the current temperature, meaning that they should melt off. This is a non-trivial shortcoming and requires manual inspection to verify correctness of constructions with respect to \(\tau\)-stability at all phases.

• WebTAS(beta) supports two slight variations of the model, and the version that matches that defined in the DNA31 submission is called 'version 2' in WebTAS(beta) (and the scripts included in this package).

• WebTAS(beta) can load and run BlockTAM systems, but does not yet support creation, editing, or saving of BlockTAM systems.

In the downloadable package linked below is a Python script allowing one to draw a path and save it as a BlockTAM system utilizing the universal shape builder construction, which can then be simulated in WebTAS. The following image shows the path drawing software on the right and the simulated system to its left.

An example shape drawn in the BlockTAM Path Drawer (right) and them simulated in WebTAS (left)

Downloads

Examples of systems cited above, as well as scripts for generating universal number encoders, universal shape builders, and the temperature sequences used to control them can be downloaded using the following link:

BlockTAM_examples. (Clicking on the most recent, bottommost link will download a zip file containing the files.)