Kinetic Tile Assembly Model (kTAM)

In reality, DNA tile self-assembly is a more complicated process than that modeled by the aTAM, and therefore a different model is required for a realistic simulation of the physical process of self-assembling DNA tiles. Whereas the aTAM is a great model for studying the capabilities and limitations of tile assembly, and for programming tile sets to understand issues related to computation and geometry, the kinetic Tile Assembly Model (kTAM) was developed as a more physically realistic model for laboratory settings, and considers the reversible nature of self-assembly, factoring in the rates of association and dissociation of basic molecular elements (so-called monomers, or tiles) within the original framework provided by the aTAM. The kTAM describes the dynamics of assembly according to a set of reversible chemical reactions: A tile can attach to an assembly anywhere that it makes even a weak bond, and any tile can dissociate from the assembly at a rate dependent on the total strength with which it adheres to the assembly. In this section, we first give a more formal definition of the kTAM, then describe the types of errors that it captures, and then discuss several results which have successfully demonstrated methods for reducing those errors. Techniques such as those discussed below have been responsible for a rapid and steady decline in the frequency of errors seen in laboratory experiments, plummeting from error rates of 10% per tile in 2004 to only 0.13% by 2009 , and continuing to shrink.

Model Definition

In the kTAM , a monomer tile can be added to the assembly with some association (forward) rate, or removed from the assembly with some dissociation (reverse) rate. Similar to the aTAM, only the singleton tiles are allowed to attach to, and in this case detach from, a seeded assembly. These rates are denoted by \(r_f\) and \(r_{r,b}\), respectively. At every available site on the perimeter of an assembly (i.e. the frontier), every possible monomer tile can associate to the assembly, regardless of whether the monomer is correct or not (i.e. whether or not the glues match). The forward rate depends only on the monomer tile concentration, \([monomer]\)\[\begin{eqnarray*} r_f = k_f[monomer] = k_f e^{-G_{mc}} \end{eqnarray*}\]

where \(G_{mc} > 0\) is the non-dimensional entropic cost of associating to an assembly. In the kTAM, for simplicity it is assumed that tile concentrations remain constant at \([monomer] = e^{-G_{mc}}\). Therefore, since the forward rate constant \(k_f\) is a constant, the entire forward rate \(r_f\) is also constant.

The reverse rate is dependent upon the binding strength b of the tile to the assembly, and in fact the relationship is exponential\[\begin{eqnarray*} r_{r,b} = k_{r,b} = k_f e^{-bG_{se}} \end{eqnarray*}\]

where \(G_{se}\) is the non-dimensional free energy cost of breaking a single bond and b is the number of "single-strength" bonds the tile has made.

The kTAM's equivalent to the aTAM's temperature \(\tau\) parameter is the ratio of the concentration of the tiles to the strength of their individual bonds, or \(G_{mc}/G_{se}\). As a simplifying assumption, the tile concentrations are considered to remain constant during assembly (despite the fact that singleton tiles will be transitioning from freely floating individual tiles to being attached to growing assemblies), which in turn causes the temperature parameter to remain constant. (It should be noted that despite this and other simplifying assumptions, the kTAM does in fact provide a quite accurate model of the systems observed in laboratory settings.) Because the kTAM allows for the binding of tiles whether or not their glues correctly match those on the boundary of a growing assembly, bindings which would be considered errors in the aTAM are possible. By lowering the ratio of \(G_{mc}/G_{se}\), which is intuitively similar to lowering the temperature \(\tau\) threshold in the aTAM, assembly happens more quickly but is more error prone. If the number of correct bonds that a tile has with an assembly, b, is less than \(\tau\), then a tile is more likely to detach than to attach.

Because the kTAM accurately models the behavior of DNA based tile self-assembly in the laboratory, most especially the common types of errors observed, it has provided an excellent foundation for work in error prevention and correction.