Tile Assembly Model


The (abstract) Tile Assembly Model (TAM) was introduced by Eric Winfree in his 1998 Ph.D. thesis.  It is a mathematical model of laboratory-based nanoscale self-assembly.  In the TAM, molecules are represented as un-rotatable, but translatable two-dimensional square 'tiles', each side of which having a particular 'glue label' and 'strength' associated with it.  Two tiles that are placed next to each other interact if the glue labels on their abutting sides match, and they bind if the strength on their abutting sides match, and the sum of such matching edges is at least a certain 'temperature.'

Essentially, the TAM is an abstract way of representing systems where sets of molecules autonomously coalesce to form structures.  In the TAM, each of the molecules can be viewed as a square tile, and new tiles can attach to an existing group of tiles only in positions where they have enough matching edges (where 'enough' is determined by the temperature value and the strength of the edges).

 

Terminology

Tile Assembly Systems

In the TAM, a tile assembly system is the complete specification for a collection of tiles which can potentially assemble into some shape or set of shapes.  The definition of a tile assembly system consists of three parts: a tile set, a seed assembly, and a temperature.

Tile Type

A tile type is the complete definition of a two-dimensional square known as a tile.  This consists of both the glue labels (which are simply sets of text characters, or strings) and strengths (integer values greater than or equal to 0) for each of the four sides (north, south, east, and west).  Additionally, it is typical to also assign a label or color to the entire tile type.

Sample Tile

Figure 1

Figure 1 shows an example of a graphical representation of a tile type.  This tile type has the label 'Foo', and the edges are defined as follows:

Tile Set

A tile set is simply a collection of distinct tile types.

Tile Assembly

A tile assembly, or simply an assembly, is a collection of tiles that have bound together.  It is specified as a list of positions and the tile types of the tiles found at each of those positions.  Note that a tile set is merely the collection of different types of tiles.  In a tile assembly, an arbitrary number of tiles of each type in the tile set may appear.

Seed Assembly

A seed assembly is an assembly that is explicitly defined and manually created, and then from which an assembly sequence begins.  In other words, it is the initial 'seed' from which an assembly grows.

Binding Strength

The binding strength of two tiles at their adjacent edges is equal to the strength of each of those edges if the glue labels and strengths for those edges are identical  (not the sum of the strengths, but merely the strength value of one of those edges since they are equal).  Otherwise, if either the glue labels or strengths differ, or the edges are not adjacent, the binding strength is 0.

Temperature

The temperature parameter (commonly referred to as 'tau') of a tile assembly system is simply an integer number greater than 0.  This number specifies the minimum value of the sum of the binding strengths of a tile with which it can bind to the assembly.  (Note that the most common temperature setting is 2, and therefore 2 is the default setting for ISU TAS.)

Assembly Sequence

An ordered series of steps whereby, during each step, a single tile is added to an assembly.

Frontier Location

A location which is adjacent to one or more tiles in an assembly, such that the sum of the strengths of the adjacent edges of those tiles is greater than or equal to the temperature value.  Intuitively, a fronteir location is a location in which a tile could possibly attach to the assembly (assuming there is a tile type that matches the adjacent edges with enough strength).

Frontier

The complete set of frontier locations for a given assembly.

Terminal Assembly

An assembly with either no frontier locations or where there are no tile types that can bind in any of the frontier locations.  A terminal assembly is one which cannot grow any larger because no more tiles can attach.

 

TAM Examples

Figure 2 shows a small sample tile set .

 

Seed Tile  Top tile  Right1 tile  Right2 tile  Foo1 tile  Foo2 tile  Foo3 tile

Figure 2

 

Say we have a tile assembly system which includes the above tile set, a seed assembly consisting of a single tile of the SEED tile type above, and a temperature of 2.  Figure 3 represents one possible assembly sequence that could occur for the first two tile additions.

 

       Seed       Stage 1                    Stage 2                                 Stage 3

Figure 3

 

In Step 1, a tile of type 'Top' was able to bind because its south label ('sy') matches the north label of the 'SEED' tile, their strengths match (they are both 2), and the sum of the binding strengths is 2 which is greater than or equal to the temperature parameter of 2 specified above.

In Step 2, a tile of type 'Right1' could bind because the labels 'sx' match and the strength is again equal to 2.

Note that the order in which Steps 1 and 2 occurred is arbitrary in this case, and is determined randomly.

In Step 3, a tile of type 'Right2' binds to the east side of the 'Right1' tile because of the matching strength 2 bonds labeled 'rx'.

Given the assembly that exists in Stage 3, and labeling the position of the 'SEED' tile as location (0,0), we'll now consider a few different cases: